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Lecture Notes in Monographs Editorial Board
Beig, Vienna, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zftrich, Switzerland R. L. Jaffe, Cambridge, MA, USA R.
Kippenhahn, G6ttingen, Germany Ojima, Kyoto, Japan H. A. WeidenmillIer, Heidelberg, Germany J. Wess, MiInchen, Germany J. Zittartz, K61n, Germany R.
I.
Managing Editor W.
Beiglb6ck
Assisted by Ms. Monika Eisendcher c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse 17, D-69121 Heidelberg, Germany
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technique
of
jean Daillant Alain Gibaud
X-Ray and Neutron Reflectivity: Principles and Applications
N
Q] -
-
04’
Springer
Author
Jean Daillant
Physique de I’Etat Saclay
Service de CEA
Condens6
F-9-ii9i Gif-sur-Yvette Cedex, France Alain Gibaud Laboratoire de
Physique de FEW Condens6,
UPRES A 6o87
Universit6 du Maine, Facult6 des sciences F-72o85 Le Mans Cedex 9, France
Library of Congress Cataloging-in- Publication Data. Die Deutsche Bibliothek
-
CIP-Einheitsaufnahme
and neutron reflectivity : principles and applications / Jean Daillant ; Alain Gibaud (ed.). Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris Singapore ; Tokyo Springer, 1999 (Lecture notes in physics : N.s. M, Monographs 58) ISBN 3-540-66195-6
X-ray
-
ISSN 0940-7677
(Lecture Notes in Physics. Monographs)
ISBN 3-540-66195-6
Springer-Verlag Berlin Heidelberg New York
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Foreword
The reflection of x-rays and neutrons from surfaces has existed as an experimental technique for almost fifty years. Nevertheless, it is only in the last decade that these methods have become
enormously popular
as
probes of
surfaces and interfaces. This appears to be due to the convergence of several different circumstances. These include the availability of more intense neutron and x-ray
of
magnitude
sources
(so
that
reflectivity
can
be measured
and the much weaker surface diffuse
detail);
over
scattering
many orders
can now
also be
growing importance of thin films and multilayers in both technology and basic research; the realization of the important role which roughness plays in the properties of surfaces and interfaces; and finally the development of statistical models to characterize the topology of roughness, its dependence on growth processes and its characterization from surface scattering experiments. The ability of x-rays and neutro4s to study surfaces over four to five orders of magnitude in length scale regardless of their environment, temperature, pressure, etc., and also their ability to probe buried interfaces often makes these probes the preferred choice for obtaining global statistical information about the microstructure of surfaces, often in a complementary manner to the local imaging microscopy techniques, This is witnessed by the veritable explosion of such studies in the published literature over the last few years. Thus these lectures will provide a useful resource for students and researchers alike, covering as they do in considerable detail most aspects of surface x-ray and neutron scattering from the basic interactions through the formal theories of scattering and finally to specific applications. studied in
some
the
weakly with surfaces the kinematic theories that so simple weakly enough general of scattering are good enough approximations to describe’the scattering. As most of us now appre, iate, this is not always true, e.g. when the reflection is close to being total, or in the neighborhood of strong Bragg reflections (e.g. from multilayers). This necessitates the need for the full dynamical. theory (which for specular reflectivity is fortunately available from the theory of optics) or for higher-order approximations, such as the distorted wave Born approximation to describe strong off-specular scattering. All these methods are discussed in detail in these lectures, as are also the ways in which the magnetic interaction between neutrons and magnetic moments can yield informaIt is often assumed that neutrons and x-rays interact
and in
interact
I
VI
tion the this
Foreword
on the magnetization densities of thin films and multilayers. I commend organizers for having organized a group of expert lecturers to present subject in a detailed but clear fa..shiou, as the importance of the subject
deserves.
S. K. Sinha
Advanced Photon Source
Argonne
National
Laboratory
December 1998
Contents
Part I.
1
Principles
Interaction of X-rays (and Neutrons) with FranVois de Bergevin
1.1
Introduction
1.2
Generalities and Definitions 1.2.1
Conventions
..................................
4
1.2.4
Scattering
........................................
and Flux
a
.............................
L,ength and Cross-Sections
1.2.5 The Use of Green
in
Functions
9
............................
11
Field
...............................................
16
Optical
an
Object
to the
..........................................
Theorem and Its Extensions
1.3.3 The Extinction
.................
Approximation Stronger
and the Born
Length
1.3.4 When the Interaction Becomes
X-Rays
Electromagnetic Propagation
13
1.3.1 Introduction
1.4
1.4.1
General Considerations
1.4.2
Classical
by
Description:
Free Electron
23 24
Scattering Description: by the Electrons of an Atom, Rayleigh Scattering 1.4.4 Quantum Description: a General Expression for Scattering and Absorption 1.4.5 Quantum Description: Elastic and Compton Scattering 1.4.6 Resonances: Absorption, Photoelectric Effect 1.4.7 Resonances: Dispersion and Anomalous Scattering 1.4.8 ’Resonances: Dispersion Relations X-Rays: Anisotropic Scattering
1.A
31
.....
34
..............
38
.........
41
........................
43
...............................
48
..........................................
48
.......................
51
...........................
52
........................
54
Magnetic Scattering Magnetic Scattering 1.5.4 Templeton Anisotropic Scattering 1.5.5 The Effect of an Anisotropy in the Appendix: the Born Approximation Anne
28
...........................
1.5.2 Non-resonant 1.5.3
27
Thomson
..........
1.5.1 Introduction
26 26
Scattering
.....................................
1.4.3 Classical
1.5
17
.......
.................................
Thomson
16
..................
....................................................
a
7
....
Scattering by
1.3.2 The
5
....................
Green Functions: the Case of the
Medium
3
..........................................
Equation Intensity, Current
1.2.6
3
4
1.2.3
From the
.........
................................................
1.2.2 Wave
1.3
Matter
Resonant
Senienac, Fran ois
de
Alain Gibaud and Guillaume
Index of Refraction
.....
54
.........................
56
Bergevin, Jean Daillant,
Vignaud
VIII
Contents
Statistical Aspects of Wave Senienac, Jean Daillani
2
Scattering
Rough
Surfaces
.
60
....................
60
......................
61
at
Anne 2.1
Introduction
2.2
Description of RandomlyRough Surfaces
...........................
2.2.1 Introduction ’2.2.2
2.2.3 2.2.4
....................
61
Height Probability Distributions Homogeneity and Ergodicity The Gaussian Probability Distribution
.........................
61
............................
63
and Various Correlation Functions
65
.....................
.......................
Complicated Geometries: Multilayers and Volume Inhomogeneities Description of a Surface Scattering Experiment,
2.2.5 More 2.3
Coherence Domains
................................
70
....................................
72
2.3.3
Coherence Domains To What Extent Is
a
Notions
...............
74
Statistical Formulation
of the Diffraction Problem Relevant? 2.4.2
68 69
2.3.2
2.4.1
67
...................................
Statistical Formulation of the Diffraction Problem
2.4
...........
.........................................
Scattering Geometry Scattering Cross-Section
2.3.1
......
(Specular) (Diffuse) Intensity
.....................
74
.......................
79
Coherent
on
and Incoherent
Statistical Formulation of the Scattered Intensity
2.5
Under the Born
...............................
80
.................
80
...................................
83
Approximation Scattering Cross-Section
2.5.1 The Differential
Flat Surfaces
2.5.2
Ideally
2.5.3
Self-Affine
Rough
Specular Reflectivity
3
Surfaces
..............................
from Smooth and
Rough
Surfaces
83
...
87
Alain Gibaud 3.1
.............
87 87
....................................
88
Concepts 3.1.2 Fresnel Reflectivity
3.1.1
Basic
Ideally Flat Surface
........................................
The Reflected Intensity from
an
............................
96
.................................
96
3.1.3 TheTransmission Coefficient
3.1.4 The Penetration 3.2
Depth
X-Ray Reflectivity in Stratified Media
.........................
99
99 :’’*--*******’ 103 3.2.2 The Refraction Matrix for X-Ray Radiation 104 3.2.3 Reflection from a Flat Homogeneous Material
3.2.1 The Matrix Method
..................
...............
..............
3.2.4 A
Single Layer
3.2.5 Two Layers 3.3 3.4
3.A
on a
on a
...........................
105
.............................
106
Substrate
Substrate
Dynamical to Kinematical Theory Influence of the Roughness on the Matrix Coefficients Appendix: The Treatment of Roughness in Specular Reflectivity Frangois de Bergevin, Jean Daillani, Alain Gibaud
.......................
108
...........
113
and Anne Sentenac
116
From
.........................................
Contents
Rayleigh Calculation Grating Treatment of Roughness in Specular Reflectivity
IX
3.A.1 Second-Order for
a
3.A.2 The
Sinusoidal
.................................
*
.......
118
...............................
118
.........................................
121
within the DWBA
..............................
3.A.3 Simple Derivation of the Debye-Waller and Croce-N6vot Factors 4
Diffuse
Scattering
116
Daillani, Anne Senienac Differential Scattering Cross-Section 4.1.1 Propagation Equation 4.1.2 Integral Equation
Jean 4.1
.................
122
..................................
122
......................................
123
for
X-Rays
4.1.3 Derivation of the Green Functions
Using
the
Reciprocity
4.1.4 Green Function in 4.1.5
a
Green Function for
Vacuum
a
..........................
125
............................
126
Theorem
Stratified Medium
Scattering Cross-Section Approximation .................................... 4.2.1 Expression of the Differential Scattering Cross-Section 4.2.2 Example: Scattering by a Single Rough Surface Distorted-Wave Born Approximation 4.3.1 Case of a Single Rough Surface 4.1.6 Differential
4.2
4.3
...................
.....................
First Born
4.3.2 General Case of
a
127 128
130
.....
130
............
131
..........................
132
..........................
133
Stratified Medium
.....................
134
.
.........
137
..........................................
140
4.3.3 Particular Case of
a
Film
......................
4.4
Polarisation Effects
4.5
Scattering by Density Inhomogeneities 4.5.1 Density Inhomogeneities in a Multilayer 4.5.2 Density Fluctuations at a Liquid Surface Further Approximations The Scattered Intensity ....................................... 4.7.1 Expression of the Scattered Intensity
4.6 4.7
.........................
141
..................
141
.................
142
.....................................
143
.....................
4.7.2 Wave-Vector Resolution Function Revisited
........................
4.1)
146 148
...........................
150
.................
152
..............................
153
4.A
Within the Born
144
........................................
Reflectivity Appendix: the Reciprocity Theorem 4.B Appendix: Verification of the Integral Equation in the Case of the Reflection by a Thin Film on a Substrate 4.C Appendix: Interface Roughness in a Multilayer 4.8
144
Approximation
Appendix: Quantum Mechanical Approach of Born and Distorted-Wave Born Approximations ..............................
155
........................................
155
Tilo Baumbach and Peir Mikulik 4.D.1
Formal
Theory
4.D.2 Formal Kinematical Treatment
by First-order
Born
Approximation
......................
157
X
Contents
4.D.3 Formal Tteatment
by 5
Neutron
a
Distorted Wave Born
Reflectometry
.
Approximation
................
158
...................................
163
Claude Fermon, Fir6d&te Ott and Alain Menelle 5.1 Introduction
................................................
5.2
..........
5.2.2 Neutron-Matter Interaction 5.3
.............................
Non-magnetic Systems Optical Indices 5.3.2 Critical Angle for Total External Reflection .5.3.3 Determination of Scattering Lengths and Optical Indices 5.3.4 Reflection on a Homogeneous Medium Neutron Reflectivity on Magnetic Systems Reflectivity
on
.........................
5.3.1 Neutron
5.4
163
165 Schr,5dinger Equation and Neutron-Matter Interactions 5.2.1 Schr6dinger Equation ................................... 165 165
168
................................
168
...............
169
....................................
170
....................
171
.....................
172
.....................
172
.....................
174
......................................
175
5.4.1 Interaction of the Neutron with
an
Infinite
5.4.2 Solution of the 5.4.3 5.4.4
General Solution
Continuity
5.4.5 Reflection 5.5
Homogeneous Layer Schr6dinger Equation
Conditions and Matrices on a
Magnetic Dioptre
......................
176
........................
179
Non-perfect Layers, Practical Problems and Experimental Limits
.....................................
5.5.1 Interface 5.5.2
Angular
Roughness Resolution
183
....................................
185
..........................
186
..........................................
186
..........................................
186
Analysis of Experimental 5.6 The Spectrometers 5.5.3
5.6.1 Introduction 5.6.2 Time of
Flight
Data
Reflectometers
5.6.3 Monochromatic Reflectometers 5.7 5.8
187
..........................
187 188
...............................
189
5.8.1 Absolute Measurement of
Bragg
...........................
..........................................
Polymer Examples Examples on Magnetic Systems 5.8.2
183
.....................................
Peaks of
a
Multilayers
............
189
.............................
190
Magnetic
Moment
5.8.3 Measurement of the In-Plane and Out-of-Plane Rotation of Moments. Measurement of the Moment Variation
Single Layer ....................................... Hysteresis Loops Conclusion on Neutron Reflectometry in
a
5.8.4 Selective 5.9
191
..............................
193
.........................
194
Contents
Part II.
6
XI
Applications
199 Physics at Crystal Surfaces Pimpinelli 199 Surface Thermodynamics 199 6.1.1 Surface Free Energy 200 6.1.2 Step Free Energy 6.1.3 Singularities of the Surface Tension ....................... 201
Statistical
.....................
Alberto 6.1
....................................
...................................
......................................
6.1.4 Surface Stiffness
.......................................
6.1.5 Surface Chemical Potential 6.2
Surface
.............................
a Crystal Morphology 6.2.1 Adatoms, Steps and Thermal Roughness of 6.2.2 The Roughening Transition 6.2.3 Smooth and Rough Surfaces 6.2.4 Diffraction from a Rough Surface 6.2.5 Capillary Waves Surface Growth and Kinetic Roughening 6.3.1 Equilibrium with the Saturated Vapour 6.3.2 Supersaturation and Vapour Deposition 6.3.3 Nucleation on a High Symmetry Substrate
of
..............................
Scaling
.............................
206
............................
206
.......................
208
Surface
210
..................
210
.................
210
................
211
..........................
...............................................
Experiments
on
210
.......................
6.3.6 Surface-Diffusion-Limited Growth Kinetics 7
204
204
6.3.4 Kink-Limited Growth Kinetics 6.3.5
202
......
a
.......................................
6.3
201
................
212
213 215 217
Solid Surfaces
Jean-Marc Gay and Laurent Lapena 7.1
....................................
217
...............................
217
..........................
223
Experimental Techniques Reflectivity Experiments 7.1.2 Roughness Investigations with Other Experimental Tools Examples of Investigations of Solid Surfaces/Interfaces 7.2.1 Co/Glass Self-Affine Gaussian Roughness 7.1.1
7.2
-
............
...............
8
223
7.2.2 Si Homoepitaxy on Misoriented Si Substrate. Structured Roughness
226
Conclusion
229
..................................
7.3
223
.................................................
X-ray Reflectivity by Rough Multilayers
..................
232
Tilo Baumbach and Petr Mikul%*k 8.1
Introduction
8.2
Rough Multilayers Multilayers 8.2.2 Multilayers with Rough Interfaces 8.2.3 Correlation Properties of Different Interfaces Setup of X-Ray Reflectivity Experiments
Description
................................................
of
8.2.1 Ideal Planar
8.3
.............................
................................
232 234 235
.......................
235
..............
235
.......................
237
XII
Contents
Experimental Setup Experimental Scans 8.4 Specular X-Ray Reflection 8.4.1 Roughness with a Gaussian Interface 8.3.1
....................................
8.3.2
...............................
Distribution Function 8.4.2
Stepped
Surfaces
238
....
239
....................................
241
;
..................................
241
......................................
248
8.4.3 Reflection
by "Virtual Interfaces" Between Porous Layers 8.5 Non-specular X-Ray Reflection 8.5.1 Interfaces with a Gaussian Roughness Profile 8.5.2 The Main Scattering Features of Non-specular Reflection by Rough Multilayers 8.5.3 Stepped Surfaces and Interfaces 8.5.4 Non-coplanar NSXR 8.6 Interface Roughness in Surface Sensitive Diffraction Methods 8.7 X-Ray Reflection from Multilayer Gratings
.................................
249
...............................
250
..............
251
..........
254
.........................
259
...................................
262
.....
264
.....................
266
.................................
267
............................................
272
8.7.1 Theoretical Treatments 8.7.2
Discussion
Reflectivity from Rough Multilayer Gratings Appendix: Reciprocal Space Constructions for Reflectivity 8.7.3
8.A
..............
8.A.1 Reflection from Planar Surfaces and Interfaces 8.A.2 Periodic
Multilayer Space Representation of DWBA
Reflectivity
of
Liquid
275
............
275
....................................
277
...............
278
,8.A.3 Reciprocal 9
273
.......
Surfaces and Interfaces
..............
281
.......................
281
Jean Daillant
Statistical
9.1
9.1.1
Description of Liquid Capillary Waves
9.1.2
Relation to Self-Affine Surfaces
Surfaces
.......................................
..........................
Bending Rigidity Experimental Measurement of the Reflectivity of Liquid Surfaces 9.2.1 Specific Experimental Difficulties 9.2.2 Reflectivity 9.2.3 Diffuse Scattering Some Examples 9.3.1 Simple Liquids Free Surface 9.3.2 Liquid Metals 9.3.3 Surfactant Monolayers Liquid-Liquid Interfaces 9.1.3
9.2
9.3
9.4
......................................
286
...........................................
287
........................
287
...........................................
288
.....................................
290
.............................................
294
.............................
295
.........................................
296
.................................
297
.....................................
300
.........................................
305
..............................................
305
Polymer Studies
10
282
285
G?inier Reiter 10.1
Introduction
10.2
Thin
Polymer
Films
.......................................
306
Contents
...................................
310
..................................
314
..........................................
315
10.3
Polymer,Bilayer Systems
10.4
Adsorbed Polymer Layers Polymer Brushes
10.5 10.6 10.7 10.8
Xiii
...................................
319
.....................................
320
.....................................
321
Interfaces
Polymer-Metal Spreading of Polymers Dewetting of Polymers
Main Notation Used in This Book
.............................
32.5
Index .......................................................... 327
List of authors
*
Dr. T. Baumbach
0
freie
Prof. A. Gibaud
Laboratoire de
Zerst6rungsPriifverfahren, EADQ Dres-
Fraunhofer Institut
Physique de FEW
Condens6, UPRESA 6087 Universit6 du Maine Facult6 des
den
Kriigerstraoe
sciences,
22
Dresden, Germany Present address: European Synchrotron Radiation Facility BP 220, F-3.8043, Grenoble Cedex France
72085 Le Mans Cedex
D-01326
*
France
Dr. L.
Lapena CNRS, Campus de Luminy
CRMC2
case
13288 Marseille Cedex
Dr. F. de
Bergevin Laboratoire de Cristallographie associ6 I’Universit6 Joseph Fourier
9,
40
9,
913, France
Dr. A. Menelle
CNRS Bitiment F
Laboratoire L6on Brillouin CEA
des martyrs, B.P. 166 38042 Grenoble Cedex 09, France
CNRS, CEA Saclay
25
avenue
91191 Gif sur Yvette
Cedex,
France
and
European Synchrotron Radiation Facility B.P. 220, 38043 Grenoble Cedex, France 0
Physique
dens6,- Orme CEA Saclay
Dr. P. Mikulik
Laboratory
de FEW Con-
91191 Gif sur Yvette
Cedex,
France
of Science
Masaryk Uni-
Kotliiski 2 611 37
des Merisiers
of Thin Films and Nanos-
tructures
Faculty versity
Dr. J. Daillant
Service de
0
Brno, Czech Republic
Dr. F. Ott
Laboratoire L6on Brillouin CEA Dr. C. Fermon
CNRS, CEA Saclay
Service de Physique de FEW Condens6, Orme des Merisiers
91191 Gif sur Yvette
CEA Saclay 91191 Gif sur Yvette Cedex, France
Prof. A.
Cedex,
Pimpinelli
LASMEA
Universit6 Blaise Pascal Dr. J.M.
Campus
Gay Luminy,
-
Cler-
mont 2
Les C6zeaux
CRMC2 CNRS, de
France
case
13288 Marseille Cedex
9,
913
France
63177 Aubi6re
Cedex,
France
XV1
List of authors
Dr. G. Reiter
Dr. G.
Institut de Chimie des Surfaces et Interfaces
CNRS, 15 rue Jean Starcky, B.P. 2488 68057 Mulhouse, France Dr. A. Sentenac:
-LOSCM/ENSPM Universit6 de St J6r6me 13397 Marseille Cedex
20,
France
Vignaud Bretagne 4 rue Jean Zay 56100 Lorient, France Universit6 de
sud
Acknowledgement s
school on reflectivity held in Luminy, France, 13th, 1997. The editors are particularly grateful to the Universit6 du Maine (Le Mans, France), to the Direction des Sciences de la Mati6re,of the Commissariat h I’Energie Atornique (C.E.A.), to the C.N.R.S. (D6partement Sciences Physiques et Math6matiques) and to the R6gion des Pays de la Loire for their help and sponsoring of this summer school. Many thanks are esp’ecially adressed to all of those who made this meeting possible: A. Radigois from the "d6l6gation C.N.R.S. Bretagne-Pays de la.Loire" who very kindly suggested the location of the school and who helped us through the admisnistrative tasks, J. Lemoine who made a wonderful job as the school secretary, and G. Ripault for his technical support at Luminy and a perfect
This book folllows
a summer
from June 9th to June
organisation of social events. We are also indebted to Dr. D. Bonhomme for helping us during the preparation of the manuscript, to Drs. N. Cowlam and T. Waigh for reading some chapters of this book, and to the staff of the C.N.R.S. center of Luminy for their hospitality.
Introduction
In his paper entitled "On
a
New Kind of
Ray, A Preliminary Communica-
relating the discovery of x-rays, which was submitted to the Wiirzburg Physico-Medical Society on December 28, 1895, R6ntgen stated the following about the refraction and reflection of the newly discovered rays: "The question as to the reflection of the X-ray may be regarded as settled, by the experiments mentioned in the preceding paragraph, in favor of the view that no noticeable regular reflection of the rays takes place from any of the substances examined. Other experiments, which I here omit, lead to tion"
the
same
conclusion.’"
This conclusion remained
unquestioned
out that if the refractive index of to be
ought
possible, according
reflection from
a
the
substance for x-rays
to the laws of
smooth surface of
optics,
tion that the reflection of x-rays
on
a
3
was
The demonstra-
obeying the laws and others who investigated the
surface
electromagnetism was pursued by Prins role of absorption on the sharpness of the limit
of
that it
to obtain total external
since the x-rays, on entering the submedium of smaller refractive index. This
it,
air, are going into a starting point for x-ray (and neutron) reflectivity.
stance from the was
a
2
Compton pointed is less than unity, it
until in 1922
was
indeed
of total reflection and showed
consistent with the Fresnel formulae. This work
was
continued
4
using nickel films evaporated on glass. Reflection on such thin films gives rise to fringes of equal inclination (the "Kiessig fringes" in the xthin film thicknesses, now the ray literature) which allow the measurement of most important application of x-ray and neutron reflectivity. It was, however, not until 1954 that Parratt5 suggested inverting the analysis and interpreting models of an inhomox-ray reflectivity as a function of angle of incidence via method The distribution. was then applied to several surface-density geneous by Kiessig
cases
of solid
or
liquid’
that "it is at first
interfaces. Whereas Parratt noticed in his 1954 paper that any experimental surface appears smooth
surprising
that, for good reflection, a mirror surface wavelength of the radiation involved..." appeared that effects of surface roughness were important, the most
to x-rays. One
frequently
hears
must be smooth to within about
it soon
A
more
complete
citation of
one
R6ntgen’s
paper is
given
in
an
appendix
to this
introduction.
Compton Phil. Mag. 45 1121 (1923) Prins, Zeit. f, Phys, 47 479 (1928); a very interesting developments is given in the famous book by R.W. James, of the diffraction of x-rays", Bell and sons, London, 1948. H. Kiessig Ann. der Physik 10 715 and 769 (1931). L.G. Parratt, Phys. Rev. 95 359 (1954). B.C. Lu and S.A. Rice, J. Chem. Phys. 68 5558 (1978).
A.H. J.A.
4 ,
6
J. Daillant and A. Gibaud: LNPm 58, pp. XIX - XXIII, 1999 © Springer-Verlag Berlin Heidelberg 1999
account of these
"the
early optical principles
XX
Introduction.
dramatic of them being the asymmetric surface reflection known
as
Yoneda
wings 7. These Yoneda wings were subsequently interpreted as diffuse scattering of the enhanced surface field for incidence or exit angle equal to the critical angle for total external reflection. The theoretical basis for the analysis of this surface diffuse scattering was established in particular through the pioneering work of Croce et al.’ In a context where coatings, thin films and nanostructured materials are playing an increasingly important role for applications, the number of studies using x-ray or neutron reflectivity dramatically increased during the 90’s, addressing vitually all kinds of interfaces: solid or liquid surfaces, buried solid-liquid or liquid-liquid interfaces, interfaces in thin films and multilayers9. Apart from the scientific and technological demand for more and more surface characterisation, at least two factors explain this blooming of x-ray and neutron reflectivity. First, the development of neutron reflectometers (Chap. 5) has been decisive, in particular for polymer physics owing to partial deuteration (Chap. 9), and an equally important contribution of neutron reflectivity can be expected for surface magnetism. Second, the use of 2nd and 3rd generation synchrotron sources has resulted in a sophistication of the techniq ue now such that not only the thicknesses but also the morphologies and correlations within and between rough interfaces can be accurately characterised for in-plane distances ranging from atomic or molecular distances to hundreds of microns. In parallel more and more accurate methods have been developed for data analysis. This book follows in June 1997. It i’s
school
a summer
organised
on
reflectivity held
into two parts, the first
in
Luminy (France)
being devoted to and the second the discussion to of one principles examples and applications. the school and the now Organising editing book, we had in mind that an number of are now non-specialists increasing using x-ray and neutron reflectometry and that the need for fulfilled. It is also true that
a
one
proper introduction to the field
if the
of
was
not
yet
principle reflectivity experiment is has the to measure extremely simple (one just intensity of a reflected beam), the technique is in fact really demanding. An important purpose of this book is therefore also to warn the beginners of experimental problems, often related to the experimental resolution, which are not necessarily apparent but may lead to serious misinterpretations. This is done in the second part of the book where specific aspects related to the nature of the samples are treated. An equally important purpose is also to share with the reader our enthusiasm for the many beautiful recent developments in reflectivity methods, and for the physics that can be can be done with it, and to give him/her the desire to do even more beautiful experiments. 7
8 9
even
a
Yoneda, Phys. Rev. 131 2010 (1963). N6vot, B. Pardo, C. R. Acad. Sc. Paris 274 803 and 855 (1972). For a recent review see for example S. Dietrich and A. Haase, Physics Reports 260 1 (1995) and the numerous examples cited in the different chapters of this Y.
P. Croce L.
book.
XXI
Introduction
As
,strongly suggested by
by considering
new
the short historical sketch
given above,
most
of x-rays (not only for interface studies) arise potentialitiesIO related to their nature of electromagnetic
of the revolutions in the
use
book therewaves, which was so controversial in the days of R6ntgen. The fore starts with a panorama of the interaction of x-rays with matter, giving both a thorough treatment of the basic principles, and an overview of more
topics like magnetic or anisotropic scattering, not only to give a following developments but also to stimulate reflection on new experiments. Then, a rigourous presentation of the statistical aspects of wave scattering at rough surfaces is given. This point, obviously important for understanding the nature of surface scattering experiments, as well as for their interpretation, is generally ignored in the x-ray literature (this chapter has been written mainly by a researcher in optics). The basic statistical properties of surfaces are introduced first. Then an ideal scattering experiment is described, and the limitations of such a description, in particular the fact that the experimental resolution is always finite, are discussed. The finiteness of advanced
firm basis to the
the resolution, leads to the introduction of ensemble averages for the calculation of the scattered intensities and to a natural distinction between coherent
(specular, equal to the average of the scattered field) and incoherent (diffuse, related to the mean-square deviation of the scattered field) scattering. These principles are immediately illustrated within the Born approximation in orcomplications resulting from the details of electromagnetic wave with matter. These more rigorous aspects of the scattering theory are treated in Chaps. 3 and 4 for specular and diffuse scattering. The matricial theory of the reflection of light in a smooth or rough stratified medium and its consequences are treated in Chap 3. This is used in Chap 4 for the treatment of diffuse scattering. The Croce approach to the distorted-wave Born approximation (DWBA) based on the use of Green functions is mainly used. This theory is currently the most popular for data analysis and is extensively used in the second part of the book’, which is devoted to applications, in particular in Chap 8. Howreviewed. The general ever, other methods used in optics are also shortly case of a stratified medium with interface roughness or density fluctuations is discussed using this DWBA, and different dynamical effects are discussed. Then, the theoretical aspects of a finite resolution function (the experimental aspects are treated in the second part of the book) are considered, as well as their implications for reflectivity experiments. The last chapter of this first part, principles, is devoted to neutron reflectometry whose specific aspects require a separated treatment. After an introduction to neutron-matter interactions, neutron reflectivity of non-magnetic materials is presented and the characteristics of the neutron spectrometers are given. Examples follow with der to avoid all the mathematical the interaction of
an
-
10
It is our opinion that fully exploiting reflectivity experiments would lead to
the
spectroscopic capabilities of interesting developments.
most
x-rays in
XXII
Introduction
particular emphasis put on the newly developed methods of investigation magnetic multilayers using polarised neutrons. The second part of the book is devoted to examples of the physics that can be done using x-ray eoad neuti7u, i -.reflectivity. The first three chapters are related to solid surfaces and multilayers, whereas the last two chapters deal with soft condensed matter. In both cases, a statistical description of the surfaces and of their properties is given first (Chap 6 and beginning of Chap 9) and examples follow. In (Ohap 7, the complete characterisation of the roughness of a single solid surface is considered. The experimental geometry, diffractometers, resolution functions are introduced first. Then, examples are given and and. the x-ray results are compared to the results obtained using complementary techniques like transmission electron microscopy and atomic force microscopy. More complicated cases of multilayers are discussed in Chap 8. The experimental setups’ are described and examples of reflectivity studies andnon-specular scattering measurements are discussed with the aim of reviewing all the important situations that can be encountered. Examples include rough multil4yers, stepped surfaces, interfaces in porous media, the role of roughness in diffraction experiments and multilayer gratings. Examples in soft condensed matter include liquid interfaces and polymers. This is a domain where the impact of reflectivity measurements has been very large because many of the very powerful complementary techniques which can be used with solid surfaces require high vacuum, and cannot be used for the characterisation of liquid interfaces. The specific aspects of liquid interface studies (mainly using x-rays) are discussed first. Experimental setups for the study of horizontal interfaces are described, and the implications of the specific features of liquid height-height correlation functions for reflectivity experiments are described. Examples include liquid-vapour interfaces, organic films at the air-water interfaces, liquid metal surfaces, and finally buried liquid-liquid interfaces. Finally, polymers at interfaces are considered in a last chapter. This is a domain where neutron reflectivity has made an invaluable contribution, in particular owing to the transparency of many materials to neutrons and to the possibility of contrast Variation. a
of
J. Daillant and A.
Saclay and May 1999
Le
Mans,
Gibaud,
Introduction
Appendix: R5ntgen’s report
on
the
importance from
of the
medium into
reflection of x-rays.
conditions here involved
general question
"With reference to the to the
XXIII
whether the
X-rays
on
can
the other
be refracted
it is most fortunate that this
another,
hand, and or
not
subject
on
passing be investigated in still another way than with the aid of prisms. Finely divided bodies in sufficiently thick. layers scatter the incident light and allow only a little of it to pass, owing to reflection and refraction; so that if powders are as transparent to X-rays as the same substances are in mass-equal amounts of material being presupposed-it follows at once that neither refraction nor regular reflection tak.es place to any sensible degree. Experiments were tried with finely powdered rock
one
may
salt, with finely electrolytic silver-powder, and with zinc-dust, such as is used investigations. In all these cases no difference was detected between the
in chemical
transparency of the powder and that of the substance in
mass, either
by observation
photographic plate... The question as to the reflection of the X-ray may be regarded as settled, by the experiments mentioned in the preceding paragraph, in favor of the view that no noticeable regular reflection of the rays takes place from any of the substances examined. Other experiments, with the fluorescent
which I here
with the
screen or
omit, lead
to the
same
conclusion.
as at first sight opposite. I exposed to the X-rays a photographic plate which and the glass side of which was turned was protected from the light by black. paper, towards the discharge-tube giving the X-rays. The sensitive film was covered, for the most part, with polished plates of platinum, lead, zinc, and aluminum, arranged in the form of a star. On the developing negative it was seen plainly that the dark-
One observation in this connection
it
seems
should, however, be mentioned,
to prove the
ening under
the
under the other
platinum,
particularly having exerted
the lead and
the aluminum
the
zinc,
was
stronger than
action at all. It appears,
plates, therefore, that these metals reflect the rays. Since, however, other explanations of second experiment, in order to be sure, I a stronger darkening are conceivable, in a metal the film and the sensitive plates a piece of thin aluminum-foil, between placed which is opaque to ultraviolet rays, but it is very transparent to the X-rays. Since the
same
result
substantially
the metals above named is
was
again obtained,
proved.
already mentioned that powders
are
If
we
as
no
the reflection of the X-rays from
compare this fact with the observation
transparent
as
coherent masses, and with
rough surfaces behave like polished bodies with reference to the passage of the X-rays, as shown as in the last experiment, we are led to the conclusion already stated that regular reflection does not take place, but that bodies behave toward X-rays as turbid media do towards light." the further fact that bodies with
X-rays (and Neutrons)
The Interaction of
1
with Matter
n,an ois
Bergevin
de
Cristallographie
Laboratoire de
assock
Joseph Fourier, CNRS,
l’Universit6
des martyrs, B.P. 166, 38042 Grenoble Cedex 09, France F, and European Synchrotron Radiation Facility, B.P-. 220, 38043 Grenoble Cedex,
B,itiment
25
avenue.
France
Introduction
1.1 The
propagation
generally presented according to an optical properties of a medium are described by a refractive the refractive index is sufficient to predict what will
of radiation is
formalism in which the
knowledge of happen at an interface,
index. A
that is to establish the Snell- Descartes’ laws and to
calculate the Fresnel coefficients for reflection and transmission. One of the objectives in this introduction will be to link the laws of propaof radiation and in particular the refractive index, to the fundamental
gation phenomena involved
in the interaction of radiation with matter. The main of the electromagnetic spectrum is process of interaction in the visible region for least an insulator). At higher energies molecules the polarisation of the (at
with x-rays, it is generally sufficient to take into account the interactions with the atoms and at the highest x-ray energies only the electrons need be considered in the interaction process. It is the nuclei of the materials which inas
teract with
neutrons, which also have
The conventions and
defined in Sect. 1.2. In the
a
second interaction with the electrons
magnetic moment. symbols which will be
for those atoms which carry
a
same
physical quantities introduced, together with
the
of Green functions. In Sect.
properties atomic scattering and the model and the
propagation will be scattering appropriate definitions
which characterise the
revised and the different of radiation will be
used in this book will be
section the basics of wave
of
a
1.3 the link between the
continuous medium
represented by
1.4 will be devoted to the a refractive index will be established. Section will include the inelastic That matter. with radiation of interaction x-ray will be described The the and elastic scattering and absorption. scattering, as
split into
a non
resonant and
a
resonant
part. Together with the questions
dispersion relations will be absorption the when the case scattering depends on the anisotropy 1.5, given. of the material will be briefly examined with reference to magnetic and to Templeton scattering. Neutron scattering will not be presented in detail in this chapter since it will appear in Chap. 5 of this book but we shall frequently
of
resonance
and
a
discussion of the
In Sect.
refer to it.
J. Daillant and A. Gibaud: LNPm 58, pp. 3 - 59, 1999 © Springer-Verlag Berlin Heidelberg 1999
Frangois
4
In the
de
present chapter, the bold italic font will be used
expressions
or
Bergevin
and the
emphasiZed
to
define
words
sentences will be in italic.
Generalities and Definitions
1.2
Conventions
1.2.1
Two conventions
be found in the literature to describe
can
a
propagating
wave, because complex quantities are not observed and the imaginary part has an arbitrary sign.. In optics and quantum mechanics a monochromatic
plane
wave
is
generally
written
as
A
e- i(wt-k.r)
oc
which is also the notation used in neutron
tallography. On the the plane wave as,
other
hand, A
x-ray
The
imaginary part
of all
scattering, even when doing cryscrystallographers are used to writing
+i(wt-k.r).
oc
complex quantities
(1.2)
are
the
opposite
of
one
another
in these two notations. Since the observed real quantities may be calculated from imaginary numbers, it is very important to keep consistently a unique convention. The
imaginary part f"
of the atomic
scattering factor
for
exam-
in x-ray crystallography is a positive number. This is correct provided that it is remembered that the complex scattering factor (f + fl + ifll) (f is the atomic form factor, also positive) is affected by a common minus
ple, used
sign, usually
left
implicit. In optics, the opposite convention is commonly quantity is the refractive index. Its imaginary part which is associated with absorption is always positive. The number of alternative choices is increased with another convention concerning the sign of the scattering wave-vector transfer q or scattering vector, which can be written as
used and the most useful
as
q
=
k,,,
-
kin
(1.3)
q
=
kin
-
k,,c
(1.4)
or
where kin and ksc the conventions
Only ter
one
5),
are
(1.3)
the incident and scattered and
(1.2)
exception will be made,
in which convention
describes the scattered be written in all
cases
used in
as
in
(1.1)
amplitude
(except
-_
vectors. In this
will be
book, adopted.
neutrons
(Chap-
will- be used. The structure factor which in the Born
with
f (q)
the
wave
crystallography, chapter devoted to
neutrons)
approximation will therefore
as,
I p(r) eiq.’dr
(1.5)
Interaction of
I
p(r)
where
X-rays (and Neutrons) with Matter
which will be discussed below. The real
scattering density,
is the
5
part of the refractive index is generally less than 1 with x-ray radiation and the refractive index is usually written as, n
I
--
-
6
where 6 and
i
-
0
are
(1.6)
positive.
imaginary part 0, equal to Ay/47r, is essentially positive (A is the and p is the absorption coefficient, see (1.84) and section 1.4.6). Note that because the opposite convention is used, the sign of the imaginary part of n is opposit ’e in visible optics. The waves will be assumed to be monochromatic in most instances, with the temporal dependence e’.". To satisfy the international standard of units, or SI units, the electromagnetic equations will be written in the rationalised
Indeed the
wavelength
MKSA system of units. The Coulombian force in
qq’147rEo r2
with -Opo
in
Propagation trons
or
which
be
propagation
a
yo
a
presented
in
propagation of a radiation whether neupartial differential equations
in
a common
form. We will discuss first the
Electromagnetic
radiation
0, 1, 2, 3)
defined
Ao
potential obeys AA,
--
(A,, A2, A3)
01c,
_o/-to , ’
potential
a
of
(1.7)
A,
-_
and A is the 3-vector
zA
potential. The charge
(9t2
02
1:
-
Xi=-C,Y,Z
For
case
represented by
in the Lorentz gauge and away from any
(92 A, =
be
by
where (P is the scalar electric 4-vector
can
potential
4-vector
A, (v
is in this system
series of second order
a vacuum.
the --
vacuum
47rlO-’.
-
Vacuum The
obeys
x-rays,
can
2,
Equation
Wave
1.2.2
c-.
=
5-X2i
I
EOILO ’
C2
Tf, the equivalent form of (1.8) is the Schr6dinger potential
neutron of wave function
equation without
any
h2
h,9Tf
2m
at
(using the convention of quantum mechanics for the sign of 2’, as discussed above). We shall consider essentially time independent problems and only then mono chromat ic’r adi ation which has frequency w127r. The time variable disappears from the equations, through 02
1
_W2
2 _C_ Ot2
.
ha
i-
C9 t
-
hw
h2 -
2m
of the relations
2
(electromagnetic field)
(1.10)
2
(Schr6dinger equation).
(1. 11)
ko
C2
use
ko
Frangois de Bergevin
ko is the the
wave
generic
vector in
field
a vacuum
or wave
function
and hw is the energy. In both cases, writing as A, yields the Helmholtz equation,
I/1 + _A
21 ko)
A
=
(1.12)
0.
The solutions to this
equation are plane waves with the wave vector ko. optics this equation is more usually expressed in terms of the electric and magnetic fields E and H, or the electric displacement and the magnetic induction D, B rather than the vector potential A,. E is related to the potential through In
OA
grad
E
-
c9t
-cgradAo
If the gauge is so that AO--O, E reduces to radiation is monochromatic, then E
--
0A -
c9t
-(OA/c9t).
If furthermore the
(1.14)
-iwA.
free field those conditions may usually be satisfied. Therefore, E and being proportional to each other, most of the discussion subsequent to
For A
a
equation (1.12) applies to E as well. Nevertheless, in the presence of electric charges, all the properties of the electromagnetic field cannot be described with the generic field written as a scalar. These particular vector or tensor properties will be addressed when necessary.
Propagation in
a
Medium Equation
(1. 12)
still
applies in
a
modified form
when the radiation propagates in a homogeneous medium rather than vacuum. All media are inhomogeneous, at least at the atomic scale, so for
even a
homogeneity will be taken as a provisional assumption whose justification will be discussed in Sect. 1.3. We also assume the isotropy of the medium, which is not the case for all materials. In the case of the electromagnetic radiation the medium is characterised by permeabilities e and p that replace -o and po in (1.8), although p can usually be kept unchanged. Though the static magnetic susceptibility can take different values in.various materials, we are concerned here with its value at the optical frequencies and above which is not significantly different from 1-to. In a medium equation (1.12) can be written as either, the moment the
( A + k 2)
A
--
(k
0
--
nko,
n
2
C[Z1,EO,,O
=
_
E/,EO)
(1-15)
or,
( A + ko2
-
U) A
The first form shows that the
=
wave
0
(U
-
vector has
2
ko
(I
-
n
2))
(1.16)
changed by a factor n, which is Schr6dinger equation
the refractive index. The second form is similar to the
Interaction of
I
X-rays (and Neutrons) with Matter
in the presence of a potential. Indeed in the case of the Schr6dinger equation, the material can be characterised by a potential V and the equation becomes, h 2m
which is
equivalent
to the
( A + k 2) 0
again
we
may define
1
2m
a
n2
h2
Yf
_
(1.17)
0
with
previous equation, U
and
+ V
(L i 8)
VI
refractive index =
I
2
-
Ulk 0
-
-
I
_
(1.19)
VlhW.
important to realise that describing the propagation in the medium by of the wave vector by a a Helmholtz equation, with just a simple change factor n or with the input of a potential U, is really just a convenience. In reality, each atom or molecule produces its own perturbation to -the radiation and the overall result is not just a simple addition of those perturbations. It happens in most cases that the Helmholtz equation can be retained in the form indicated above. How n or U depends on the atomic or molecular scattering has to be established. Before addressing this question we have to give some further definitions for the intensity, current, and flux of the radiation, and to introduce the formalism of scattering length, cross-section and Green functions which help to handle the scattering phenomena.. It is
1.2.3
Intensity,
Current and Flux
12
The square of the modulus of the field amplitude, i.e. JA , defines the intensity of the radiation, which is used to represent either the probability of of energy a quantum of radiation in a given volume or the density
finding transported by
the radiation.
vector direction to
correct in
vacuum
measure
JA 12
is also used when combined with the
the flux
density. These
but need to be revised in
a
definitions
are
wave-
trivially
material.
given surface is the amount of radiation, measured as unit time; an energy or a number of particles, which crosses this surface per that we shall current density this is a scalar quantity. The flux density or the electromagnetic energy flux also call the flow is a vector. For instance, in flux the an elementary surface do- is energy density (flow) is designed by S; to the flow by a relation connected is then S.do-. The density of energy u The
flux
across a
which expresses the energy conservation. The amount of energy which enters the energy inside that a given closed volume must be equal to the variation of volume:
aSX Ox
+
c9sy Oy
+
as, (9Z
Ou +
-
(9 t
0.
(1.20)
Frangois de Bergevin This
equation is no longer valid when the medium is absorbing. equation (1.20) can also be written in terms of the number of particles instead of the energy; for instance this is appropriate for the case of neutrons or for electromagnetic radiation if it is quantised. The same formalism stands for the flux, the density of current and the density of particles. The dimension of the density of flux is the one of the relevant quantity (energy, number of particles or other) divided by dimension L 2T. In the case of electromagnetic radiation, the quantities E, H, D and B, can be used instead of A as discussed above and the dielectric and magnetic Note that
-
permeabilities, and y, can density is then given by,
be used to characterise the medium. The energy
-
u
For
a
plane
wave
defined
(,-E.E*
--
by
pH.H*) /4.
+
H
and the energy
i along
the unit vector
(1.21) the
wave
vector,
(1.22)
E
x
density becomes, u
The energy flow is then S
=
equal
E
Note that these formulae
x
=
-
to the
H*/2
giving
(1.23)
Poynting
vector
cs-\I oyofty JE 12 i/2.
=
u
JE 12 /2.
and S
are
(1.24)
written in terms of
complex field
quantities
whose real part represents the physical field. The complex and the real formulations differ by a factor 1/2 in the expressions of second order in
the fields. The
change
in the
wave
vector
medium has been written above
so
that if y
--
po
(1.15), u
S
-
=
This shows that the
2
nc
(1.15)
k
=
n
_-
in
going
from
a
vacuum
into
in terms of the refractive index
nko
a
n
(1.25)
V______o P/ P 0
(1.26)
1
obtain
(,O/,to//,) JE 12 /2
-
(,-oyo/y) JEJ 2k/2
n260 JE 12 /2
--
nc,-o
JE 12 Q2.
(1.27) (1.28)
flux through a surface depends on both the amplitude E refractive index of the medium. A similar expression stands for neutrons (beware, in what follows as usual neutron physics i has the opposite sign). Here the probability density p
and in
n
we
length
also
on
the
and the current wave
a
density j
of
I Tf 12
j
particles
plane
-_
wave
Tfoe
,
P
(hil 2 m) (TfgradTf
--
ik.r
I TfO 12,
=
acterised
by
a
j
depends on potential V and
amplitude
both TfO and
n
and which from
+ V
to introduce
=
The above formulae
are
--
gradT/)
is the
along
(1.29)
.
k
rti:;
V
on
(1.30)
the medium which is char-
(1.31)
hw.
--
refractive index, which is
a
-
(1.19)
V
(1.32)
ru’.;
ko gives the length of the
j
*
(hk/m) I Tf012 .
--
h2 k2/2rn
optics, it is possible
TV
the unit vector
being
,
Here too, the current
As in
considered. The
are
9
function TV. P
For
X-rays (arid Neutrons) with Matter
interaction of
I
wave
vector k. Then
(nhko/m) I Tf012 j.
(1.33) it is
isotropic. When
valid when the medium is
anisotropic the flow of energy and the current are affected. In the electro-, magnetic case the direction of the flow does not always coincide with the direction of the
wave
vector-
Exercise 1.2.1. A beam a
impinging on amplitudes
transmitted beam. The
nel formulae
absorbing.
(see
section
As assumed
Check the conservation of the
Exercise 1.2.2. Let
jpoe(ik ,x-k z). 1.2.4
3.1.2).
a surface
us
consider
a wave
Calculate the current
Scattering Length
rise to
gives
of these beams
flux,
above,
are
reflected and
given by
the Fres-
the twa media
Tf, such
not
are ’
at least for the
function
a
as an
(s) polari’s’ation. evanescent
wave
density.
and Cross-Sections
scattering object (molecule, atom, electron), fully incident wave. The object reemits part of the incident radiation. We start with the assumption that its dimensions are small compared to the wavelength so that the scattered amplitude is the same in all the directions; for an extended object instead, direction-dependent phase shifts would appear between the scattered amplitudes coming from different regions in the sample. When examining the scattered amplitude at large distances r from the object, simple arguments yield the following expression of the scattered amplitude (see also the appendix 1.A to this chapter) Let
us
consider
an
isolated
immersed in the field of
an
A,3c
--
-Ailb
e-
ikr
r
(1-34)
Frangois de Bergevin
10
spherical symmetry (k and r are scalars), amplitude Ai,, and has locally the right wave-
Indeed this function which has the
proportional to the incident length 27r/k; the decay as a function is
of the inverse of
r guarantees the conserintensity decays as the inverse of the surface of a sphere of radius r. The remaining coefficient b has the dimension of a length; this coefficient characterises the scattering power of the sample and is the so-called scattering length. The notation b is rather used in the context of neutron scattering. Here we adopt it for x-rays as well. To be fully consistent with this notation we keep, as a mere convention, the minus sign in the definition of b. With this sign, the b value is positive for neutrons with most nuclei, and also for x-ray Thomson scattering. This length can have a complex value, since the wave can undergo a phase shift during the interaction process; we shall see that in our case if the sample is not absorbing then b is nearly real. A more rigorous justification of the expression (1.34) will be given in the next section. To justify that b has the dimension of a length, we have considered the
vation of the total flux since the related
scattered flux. The ratio of this flux to the incident
(flux density,
or
current)
has the dimension of
0-scat,tot This is the so-called total
47r
=
lb 12
a
one
per unit of surface
surface and is
general,
with
an
(1-35)
scattering cross-section
extended
the
object,
to
.
The scattered flux
the whole space is then equal to the one received by a surface which would be placed normal tothe incident beam. In
equal
equal
scattering depends fi, so that b, which
in
to crscat,tot
on
the direc-
depends fi, is written b(fi). Therefore it is useful to define a cross-section for this particular direction that is called the differential scattering cross-section tion of
observation, defined by
a
unit vector
also
on
(do-IdS?) (fi) which is for
a
equal
--
lb(fi) 12
to the measured flux in the solid
unit incident flux
(Fig. 1.1).
In this
case
(1.36)
angle
dQ directed towards
the definition
(1.35)
is
fi, replaced
by O"scat,tot
-
ff lb(fi) 12
(1-37)
dS?
integration is carried out over all the directions defined by fi. Any object (atom, molecule) also absorbs some part of the incident radiation without scattering it. Therefore one has to define the so-called crosssection of absorption, Cabs, equal to the ratio of the absorbed flux to the
where the
incident flux
notation
density.
(O"scat,tot)
We have used
in
to recall that it is
total cross-section
appellation,
(1.35) a
and
(1.37)
a
somewhat
scattering cross-section;
ut,,t, is also used to define the
clumsy
indeed the sum
of the
I
Interaction of
X-rays (and Neutrons) with Matter
11
dQ M
0
k
scattering length b(fl) and of the differential scattering -ik.r The incident plane wave is Ai,, and the scattered This last expression gives a well-defined flux in the wave Ai,#(fi)/OM)e-’kOM. The scattering length and the differential cone OM, whatever the distance OM. scattering cross-section in the direction Q are respectively b(d) and lb(fi) 12
Fig.
1.1. Definition of the
cross-section
(daldfl) (ft).
cross-section
concerning
inelastic
scattering);
all the interaction processes
it is the whole relative flux
O’scat,tot + Cabs-
O’tot
(absorption,
picked
up
by
the
elastic and
object,
(1.38)
The Use of Green Functions
1.2.5
scattering amplitude b in equation (1.34) has not been introduced very rigorously and it is possible to define it more formally. The field scattered by a point like object obeys the wave equation (1.15) everywhere except at the center of the object, which is both the source and a singular point. The simplest mathematical singularity is the Dirac J function. The Green function of equation (1.15), G(r) is a solution of the equation The
(, A + k2 )
G (r)
=
J (r).
(1.39)
Physically, G(r) represents the field emitted by the source normalised to unity. More, generally, any partial derivative equation which is homogeneous in A such
as
DA(x) (here plus
D a
--
0
(1.40)
represents a sum of differential operators with constant coefficients term, and x is a scalar or a vector variable), admits Green
constant
functions G which
satisfy
DG(x)
--
J(x).
Frangois de Bergevin
12
application of Green functions is the resolution of non homogeneous partial derivative equations. For example, if G(x) is a Green function and Ao (x) is any of the solutions of the homogeneous equation, the equation A
common
DA(x) admits the
following
can
be shown
--
Ao(x)
j G(x
+
by substitution
AX) and
(1.42)
solutions
A(x) This
f (x)
=
=
x) f (x’) dx’.
(1.42)
into
j 6(x
-
-
and
of the
use
x’)f (x’)dx’,
(1.43) equation,
(1.44)
by finally applying (1.41).
diverging wave (1.34) (or the converging wave is indeed, to a certain coefficient, a Green function solution of (1.39). Due to the spherical symmetry, it is worth using the spherical coordinates r, ft (r rfi; fi is defined by the polar angles 0, 0). The differential operators yield Let
having
us, now
the
check that the
opposite sign
k)
for
--
where d, and d2
perpendicular
grad
=
ZA
__
are
fi
a Or
Idi (0, 0),
+
02
2 0
Cqr2
fi.di
--
0
r
+ r
1
,_r
+
72
d2 (0)
0)
(1.45) (1.46)
1
differential operators relative to (0, 0) and d, is a vector moment, the above expressions are sufficient since
to ft. For the
only use functions having the spherical symmetry and therefore d, and d2 van-ish;’we shall also’use these expressions for less symmetrical functions, but in such a case we shall only consider the asymptotic behavior at large values of r where the exponent of 1/r is sufficient to make d, and d2 negligible. We then have exactly
we
ikr
(zA + k 2)
-
for
0
r
r
:A
0.
(1.47)
singularity with 6(r). It is possible to integrate a sphere of radius ro centered at the (1.47), of definition the Indeed from G, the integral of (A + k 2) G(r) must origin. be equal to 1 when performed over the whole volume including the origin. This calculation is proposed in the exercise 1.2.3 and yields -47r. The Green function of the, three dimensional Helmholtz equation is then At
r
=
0,
we
must compare the
the left hand side of
inside
G (r)
1 =
47rr
eikr
(1.48)
X-rays (arid Neutrons) with Matter
Interaction of
I
It is also useful to express the Green function in
specular reflectivity calculation yields
some
problems
can
Gld,(r) The Green function of Helmholtz
help harmonic, with with the
Tk
e
one
(1.49)
in two dimensions
equation
r-’-’ decay and
an
dimension. A similar
ikr
of Bessel functions. The a
dimension. Indeed in
one
be solved in
13
asymptotic additional phase
expressed large r is yet equal to (Ir/4). be
can
form at
shift
Exercise 1.2.3. Calculate in three dimensions
J’ A origin. The second term
extremum at the
a non
As for the first term, it
(1.76)
dXM.
singular b(20) presents (1.74) is thus negligible on this condition.
and if
is of the
(1.75)
XM2 dcos20
which allows to bound the second term
in
I
d
OM
d
integral
dXM
M
dXM
This
(1.73)
slowly with 0, it clearly appears that the second term the first one multiplied by A/OM. This comes from
varies
order
same
(20) e-’k11XMdXM.
behind the
[b (20) e-ikoXM]’ Om If
b M
plane P, the function to be integrated soon as 0; the integration easily gives almost constant and equal since b remains (1.78)), This can be rigorously proved by integrating by parts:
enough
oscillates very quickly as the result (see below Eq. to its value
10,
quickly
oscillates around
towards the infinite value of the upper bound
so
zero
that
when XM tends
one
can
make the
following approximation b
It is worth
larger
Fresnel
-
0.
XM-+00
(1.77)
that to average those oscillations, the upper bound value of the ring used in integrating over the plane P should be
noting
for the radius
much
(20) e-’kOxm
zone
r
than
a
characteristic
length.
This
radius which is -of the order of
length
(AOM)1/2.
is the so-called first
Frangois de Bergevin
20
Finally the forward
scattered
A,,, (M)
amplitude becomes
i.A(O)Ap,b(O)C k0om.
--
A(0)e-’1’-00M
The forward scattered field adds to the incident field
yields A If
total field
a
we now
in M and
A(M)
A (0) e -k’0M +
(M)
(1.78)
A,;, (M)
=
A (0)
(1
+
iAp, b (0))
e
-ikoOM
(1.79)
.
consider instead of
calculation remains valid
volume density p,
a plane a thin layer of thickness dx the above provided the surface density p, is related to the
by ps
The total field for such A
(M)
a
--
layer
_-
p, dx.
(1.80)
becomes
A(O) (I
+
iAp,b(O) dx)
e-
ikoOM
(1-81)
possible to deduce the optical theorem from this relation but this will be presented later. Note that the amplitude in M is outphased by 7r/2 relative to the one scattered by a volume element; that phase difference results from the summation of amplitudes in Fresnel diffraction. It is
The Propagation of a Wave in a Homogenous Population of Scattering Objects Let us now consider the plane P as an infinitesimal small
layer
of thickness dx made of
a
medium of index
n.
The
wave
vector in the
point 0 is located at the entrance of the layer, wave which has crossed the. thickness dx in the medium of index n amplitude at the point M given by medium is nko. If the
A
The
me-inhodx
e
-iko(OM-dx)
I_A2 p,b(O)/27r
Equations (1.82) and (1.81) plane in Fig. 1.3. As shown -
A(O) (I
-
i
(n
1) ko dx) e-’kOom.
plane
has
an
(1.82)
comparison of (1.82) with (1.81) shows that n-_
(n
, ’:z
a
1)
modifies the absolute
--
I
-
(27r/k 02) p, b (0).
(1-83)
schematically represented in the, complex figure, the imaginary part of b(O) or value of thefield amplitude in M, whereas the are
in this
real part modifies its
phase. Equation (1.83) links the scattering by elementary objects to the propagation in the medium which is considered to be continuous. It is an extension of the optical theorem. Indeed the imaginary part of n, i.e. 0, describes the attenuation of the radiation in the medium and 20ko is the absorption coefficient p:
JA(M) 12
-
dx-+O
JA(0) 12 (1
-
2#kf) dx)
=
JA(0) 12 (1
-
Cattenp,
dx),
(1.84)
Interaction of
I
X-rays (and Neutrons)
with Matter
21
Pmaginary
,A >
A
(0)
Real
complex plane. Up of the incident iAp,b(O)A(O) in the first calculation, field A(O) and of an infinitesimal field dA -i (n and dA 1) ko dxA0 in the second one. The component of the field dA 1 ) is turned by 7r/2 from the incident associated with the real part of b(O) or of (n field. This produces a phase shift of the total field, On the other hand, the imaginary I) decreases the amplitude of the total field part of b(o) or of (n
Fig.
1.3.
to the
e-"’00m,
factor
of the field in the
amplitude
of the
Representation
common
the total
field A(M)
is the
sum
=
=
-
-
-
where 0-atten is the attenuation cross-section of these objects in this medium. It is "almost" the optical theorem (1 .69). The "almost" means rep lacing atot,
the total cross-section of the isolated section in this
particular medium,
objects, by their attenuation
cross-
0atten.
approxielectromagnetic field. Indeed, only the forward scattering, which is usually independent of polarisation and conserving it, is retained. Beyond the above, approximations, and the scalar and one must take into account all the scattering direction’s behaviors. different vector fields display All the derivations above consider the field
(XM
mations made here
are
(1.82)
(1.82)
on
now
discuss the
the
on
and from
is valid if
(n and
scalar. Under the
approximations equality of the amplitudes calthe scattering (1.81). On the one
We must
argument relies
made. The
culated from the index hand
oo) they
Approximations
About the
which
-+
as a
be extended to the
can
the other hand
(1.81)
-
1) ko
dx < I
holds if
OM >> A. We
are
going
to show that in the
two conditions
can
case
of
simultaneously
hold
a
medium of finite thickness
under
some
restrictions
x
on n or
these b
(0).
An arbitrary thickness x of the material may be divided into thin layers of thickness *dx. Let 0 and M be the points taken at the entrance and at the exit of Let
0
layer j, such us
by (1.82)
into
(1-81)
as
dx
show that the as
far
as
I
=
OM
(see Fig. 1.4).
amplitude n
-
I
except in the global
at M
can
be deduced from the
one
at
I koOM < 1. Since OM does not appear phase factor, the expression of the total field
Frangois
22
Fig.
1.4. The
if
Bergevin
point M
is located at the border of the two layers (j, j+j) of the that the condition L, >> OM > A (see text) is satisfied. Note OM > A then NM > A for nearly all N. Then the amplitude at M only
material. We that
de
assume
from the layer j, and is given by (1.81) (with p, the surface density of the layer). Since L, > OM, (n I)ko dx is infinitesimally small, the approximation (1.82) does apply, and the material has an index given by (1.83) comes
-
holds whatever the value of OM
dx. Yet, for any point N even if OM points 0 and M, the condition NM > A has to be verified. Although there are some points N very close to M which do not verify NM > A, most of the points of the layer are at a distance from M --
located between the
larger
than A since the initial condition
was
OM > A.
In order for the
amplitude at M to be given by (1-81), it is also necessary for the back scattering coming from the layers j + I located behind the point M to be negligible. The different points of that layer scatter towards M with different phase shifts. It is possible to show that the ratio of the sum of scattered amplitudes in the backward direction to the one in the forward direction by layer j is of the order of A/OM. Therefore the condition OM > A is sufficient for equation (1.81) to be valid. Equations (1.81) and (1.82) are simultaneously satisfied if OM > A and In 11 kOOM < 1. The combination of these two inequalities yields -
A < OM
A is The times ten For neutrons this ratio is about larger. thus well satisfied. If V,, is of the order of A’ 2
< L A is of the order of-
inequality Vaver last inequality can
,
the volume
10’ V,,
(10’0
L.2A
Va for
involved in the
neutrons).
The
materials.
reasonably homogeneous wave propagation according to the index given by (1-83) is with the continuous medium field equations consequently valid. be easily checked in
Provided these conditions ’are satisfied, the
1.3.3
The Extinction
The condition
(1.85) (L,
Length
>
and the Born
A)’shows
role in the evaluation of the
Approximation
that the extinction
strength
length plays
of the interaction of
a
a
major
radiation with
undergo ma phase shift of exactly one The substantial. becomes then results may terial; the scattering produced been which has material the of thickness the when different be qualitatively crossed is smaller or bigger than L,. For x-rays of energy 10 keV, the extincand it is one order I I < 10 tion length is of the order of a micron (I n for neutrons. of magnitude larger The approximation which has been made to relate n with b is connected first Born approximation. We have used a single scattering to produce the to the plane wave propagating in the medium. In addition the extinction length allows us to decide whether the Born approximation is valid for a given situation. When L, > A, the criterion is that the path travelled in the volume of the material giving rise to a coherent scattering must be less than L,. The kinematical theory of diffraction by crystals (equivalent to the Born
matter.-When the radiation has travelled a
measurable
a
distance L, it
begins
to
radian because of the crossed-
-
Frangois de Bergevin
24
approximation) is commonly used because the volume of the perfect crystal (coherently scattering) is often smaller than one micron cube. The property expressed in equation (1.85), which tells us that the extinction length is much larger
than the
wavelength,
also, presents beneficial effects for the physics of even if the kinematical
x-rays and neutrons. It is associated with the fact that
theory
is
longer valid, in perfect crystals, the dynamical theory remains optics, where this condition is not valid, the diffraction equations are most often not exactly solvable. In the domain of reflection in grazing incidence on a surface, the extinction length plays a major role. First it is related to the critical angle of total external reflection, discussed in chapter 3. Indeed the following relation stands no
calculable. In visible
I / I q,
I
--
L, sin 0, / 2
(1
+
n)
;z
L, sin 0, /4,
(1-87)
where I q, I is the scattering wave vector transfer corresponding to the critical specular reflection at the critical angle 0,. The left-hand side term represents (up to a factor I / 47r) a sort of wavelength perpendicular to the surface, and the right-hand side term (up to a factor 1 / 4) the extinction length projected on the perpendicular axis. The quasi equality of these two lengths is the sign that at the critical angle, the Born approximation is no longer valid. For less shallow angles, the perpendicular wavelength becomes smaller than the perpendicular extinction length and therefore the reflectivity becomes weak and calculable in this approximation. In the case of a rough surface, one must also compare the extinction length to the characteristic lengths of its waviness. If the waviness is
losses in
1.3.4
longer
or
shorter than the extinction
reflectivity and the scattering
are
different
When the Interaction Becomes
It. is -useful to know,
even
though
this does not
length,
the
(see appendix 3.A).
Stronger apply
to neutrons or.x-rays, the
kind of
propagation which arises when the interaction becomes stronger. In such: a. case, the representation by a -continuous medium can still be retained, but the -value of the index is
longer the one given above. In particular easily without making the supposition that the scattering length b(20) is independent of the 20 scattering direction. The formation of the index now implies that multiple scattering will be produced in all directions and not only in the forward direction. Also the scalar and vector fields do not have the same properties, since for the latter b depends on 20 because of polarisation (but we know how to take it into account provided no
the index cannot be calculated
that there is
no
scalar
other
anisotropy).
field, when b is independent of the angle, the calculation that leads to the amplitude (1.78) scattered by a plane is exact, even if OM is not much larger than A. Once the integration is made in the planes perpendicular to the propagation, it is then possible to work in one single dimension. Nevertheless the discussion which uses the decomposition of the For
a
Interaction of
I
X-rays (and Neutrons)
with Matter
25
layers (Fig. 1.4) must be revisited essentially because it is no longer possible to neglect the back scattering at M coming from the other layers which are located behind the point M. A calculation is proposed in exercise 1.3.3. As indicated in the introduction 1.3.1, it yields material in
n2 For vector fields
have found
an
the molecular
=
(the
1
-
2A 2p,b/27r
=
1
(47r/k2) p, b. 0
-
(1.88)
electromagnetic field), Clausius and Mossoti polarisability of the medium to similar expression due to Lorenz and Lorentz,
of the
case
which links the static
expression polarisability.
A
gives the refractive ind ex. Usi ng our notations, this is written Lorentz classical radius Of the electron, defined in section 1.4.2),
(n2
/(n2+ 2)
(47r/3) k2PVre. 0
(r,
is the
(1-89)
apply when the extinction length and the wavelength are of homogeneity must be verified at scales shorter than the wavelength. If it is not the case the propagation may be no longer possible; it is the phenomenon of localisation. These formulae the
order. The
same
Exercise 1.3. 1. A scalar
plane
wave, with the
interface
wave
ko, enters a ko. By dividing
-vector
0 with
the
angle planar making layers parallel to the interface, calculate the scattered amplitude at any point in the medium, as shown in (1.78). Find the direction of equiphase planes of the total amplitude and compare to Snell- Descartes’s law. For which values of 0, is the approximation improper ? Hint. One can show that the scattered amplitude at a point located at the back of an angled layer is given by expression (1.78) divided by sin 0. medium
through
a
the medium in
Exercise 1.3.2. In the
same
configuration
as
the
one
of the
previous exercise,
assuming b(20) constant, find with the same method the amplitude reflected by the interface. Compare with the exact Fresnel expression given in and
chapter 3, section
3.1. In the section 3.3 in the
imation is discussed
as
chapter the Born approxamplitude calculated negligible in the discussion of
same
in this exercise. Note that the
here is the bascliscattered one, considered as the approximations at the end of section 1.3.2. Hint. The
expressions
for the scattered
amplitudes
reflected and incident .
Notice. If the 0
vectors).
big enough to allow b(20) :A b(O), this Fresnel reflectivity expression is not exact.
angle
shows that the scalar Exercise 1.3.3. In
wave
symmetrical Only b may change angle between the
at two
points with respect to an infinitesimal layer are the same. from b(O) in one case to b(20) in the other (20 is the
a one
which receives the scalar
is
calculation
medium, the dx element located at x’, amplitude A, scatters the wave in the two opposite
dimensional
Frangois
26
de
Bergevin
directions
A7.7Gld(x Gld is the the
at
-
XI)
dimension Green
one
=
.-Apblddxe i1ko(x-x’)1
function,
a
q
constant coefficient and
pbld
power which in ger eral Is Imaginary. Find the relation between pbld and the refractive index in this medium
of
density
one
scattering
dimension.
Hints. One
can
negative from the
vacuum
by
and becomes
the
--
x
A
can
-e-""ox is incident
0 between the
--
wave
A(x)
--
vacuum
Aoe-’kox
at
comes
A’e-"’k0-T in the medium. The field
=
A (x) +
10"o A(x’)77Gld(X
the transmission at the interface A’
by
transmission One
positive
A(x)
x.
can
A’(x) -
interface at
an
be written into two ways: integral equation of the scattering
in the medium -
consider
and the medium at
x
coefficient,
which is
notice that the
at the
wave
(see
origin
--
-
x) dx’;
tA0 where
2/ (n + 1) (Chapter 3,
t is the Fresnel
section 3. 1,
Eqs. (3.18),
scattering is composed of two terms. The one in disappearance (so-called extinction) of the
of the
the extinction theorem
[3]).
X-Rays
1.4
General Considerations
1.4.1
electromagnetic radiation interacts principally with the electrons, and weakly with atomic nuclei (the ratio of the amplitudes is in the inverse of masses). The interaction is essentially between the electric field and the charge, but a much weaker interaction is also manifest between the electromagnetic field and the spin, or its associated magnetic moment. A photon which meets an atom can undergo one of the three following
The
very
events:
scattering, with no change in energy; scattering: part of the energy is transferred to the atom, the most frequently with the ejection of an electron (the so-called Compton effect); however it may happen that the lost energy brings the atom in an -
-
elastic
inelastic
excited state, without any ionisation (Raman effect); absorption: all the energy is transferred to the atom and the photon vanishes. Another photon can be emitted, but with a lower energy: this is -
the so-called
fluorescence.
These mechanisms James
[4]
found in
is
particularly
[5]).
described in many text books; the one of R. W. complete (except for the Raman effect which can be
are
Interaction of
I
X-rays (and Neutrons) with Matter
27
"To give an intuitive image, we shall begin with the classical mechanics theory which simply provides an exact result for the scattering by a free electron (Thomson scattering). When the electron is bound, this theory is still convenient enough. However the Compton scattering cannot be described by this classical theory. Also this theory does not describe correctly the motion of the electrons in the atom. Therefore we shall also review all the following processes in the frame of the quantum theory, i.e.: -
the elastic and inelastic
or an
electron bound to
atomic resonance-, the photo-electric -
-
the
dispersion
an
scattering (mainly Compton), -for
a
free electron
atom, when the radiation energy is well above the
absorption by an atom; brought to the elastic scattering by
correction
the atomic
resonance.
Finally we shall discuss the general properties of dispersion which are independent of a particular interaction or radiation. One can show that the real and imaginary parts of the scattering are linked by the Kramers-Kronig relations which are extremely general and probe the response of nearly every system to some kind of excitation. The origin of these properties lies in the thermodynamical irreversibility that can be introduced through the principle of causality. Classical
1.4.2
Thomson
Description: Scattering by
Free Electron
a
scattering by a free electron is simple and presents the main characters scattering by an atom. We shall start with this case. The electron undergoes an acceleration, which is due to the force exerted
The
of the
by
the incident electric field
Ei,,(t) Let
z
be the electron
Eo iwt
-
(1.90)
position and (-e) its charge, then Tni
--
iwt (-e) Eoe
The electron exhibits oscillations of small
amplitude, producing
a
localised
current
j (r,t)
e)
i
(1.92)
(r)
(_e)2 Ein (1) iwm The radiation of that
vibrating current,
J (r).
similar to
large
distances
(kr
>
E,,
a
dipole antenna,
(1.63)
discussed in section 1.2.6. From the formulae
(1.55),
and
has been
we
have at
1),’
hl: 00
[Ei,,
(Ei,,.r)
r
r2
I
(-e) 2e-ikr ,
-
47rEOMC2,r
’
(1-93)
Frangois de Bergevin
28
What is measured is the
projection of the field on some polarisation direcand iin is the unit vector which describes given by the unit vector the incident polarisation. These vectors are chosen so that ;&in is parallel or antiparallel to EO and’ s, normal to r (see Fig. 1.5) tion
Ein
-
(Ein-;40; in
and
rJ sc
In these conditions of polarisation the definition of the can
be
adapted
as
we
(1.94)
0.
scattering length (1.34)
follows
-Ein - in then
-_
e-ikr
b(; ,,c,i i,,),
(1.95)
have
b(
i,,c, iin)
::::::
re’ sc - iin
3
where r, is the Lorentz classical radius of the electron with charge 2.818 x 10-15 M).2 The charge of the e 2/47r6o rnC2 e and mass m., (r, -
-
electron appears twice, first in the movement and then for the emission of the radiation. Thus it appears as a square and b does not depend on its
opposite to the incident one because of (by convention, a positive value of b corresponds to such a sign reversal). If the ingoing polarisation is normal or parallel to the plane of scattering, the outgoing one has the same orientation. These polarisation modes are called (s)-(s) (or (o-)-(o-)) when perpendicular to the plane of scattering and (p)-(p) (or (7r)-(,7r)) when parallel. The polarisation The
sign.
scatteredfield
is however
its relation with the current
factor of the
scattering length
the latter. The process that
scattering.
1.4.3
case
and
cos
20
(Fig. 1.5)
in
have described is the so-called Thomson
.
Classical
by
is I in the former
we
Description: Thomson Scattering an Atom, Rayleigh Scattering
the Electrons of
scattering is exact, even for the bound frequency of the x-rays is large compared to the characteristic atomic frequencies. Nevertheless it is necessary to take into account both the number of electrons and their position in the electronic cloud when calculating the scattering from an atom. Every point of the electronic cloud is considered to scatter independently from the others and the scattered amplitudes add coherently. As in any interference calculation within the Born approximation (see the Appendix LA), justified whenever The
simple result
electrons of
an
of the Thomson
atom,
as
far
as
the
A system of units which is often used to describe microscopic 2 2 Gauss system. In this system we have r, e /mc =
phenomena
is the
I
Interaction of
X-rays (and Neutrons) with Matter
(n e in
29
2,0
(a) 6sc in
Wf:o
2,0
2
(b) Fig.
(a)
1.5.
Directions of incident and scattered
mode and
factor
is
(b)
respectively
scattering
the
the
(p)-(p) or(r)-(7r)
the total atomic
(a)
the
(s)-(s) or(a)-
amplitude polarisation
f (q)
(q),
--
scattering length b"t by
density p(r)
the Fourier transform of the electron
bat
for
I and cos20
weak,- one obtains
is
polarisations
mode, The associated
f
--
p
(r) eiq.rd(r)
(1.97)
the definition of q in. (1.25)). The quantity f (q) is called the atomic scattering factor or the atomic form factor. The integral of p(r) over
(see
all
r
values must be
equal
to the number of electrons in the atom:
f (0)
__
Z.
(1.98)
.
explanation to support the validity of this interference calculation. The justification comes from the alternative quantum calculation which gives the same result. The assumption that the frequency of the radiation is greater than the atomic frequency may not be valid especially for the inner electronic shells. The model can be improved by introducing the binding of the electron to the atom which is modelled by a restoring force of stiffness K -and a damping coefficient -/. The damping is the result of the radiation which is emitted by the electron, or of the energy transferred to other electrons. The equation of motion (1.91), still written for a single electron, now becomes
There is
no
safe
mi + One looks for
a
^/i
+
NZ
solution of the kind
(-MW
2
+
i’YW
+
-_
(-e)Eoe
(e"t)
2)Z
MW 0
iWt
which must satisfy =
(-e)E0eiWt’
(1-100)
Frangois de Bergevin
30
where
tc/m
(-e)i
-wo. The current
j (r, t)
is then
iw(-e)2 Ei,, (t) J (r) ,rn (W2 W2) 0
:--
(1.101)
-
As shown for the Thomson
scattering above,
this
yields the following
scat-
tering length 2
b
We shall
when
now
only
one
r,
W2
electron and
more
heavy
just
W20
i7W/M
e,,. ein
(1.102)
-
is modified for different
expression
are
considered
general. Actually it happens w with wo. For high energy
although that W,
expression.
Wo
energies this dis-
>>-Y/m
x-rays and not too
>>wo. Within these
or even w
(L 10 2) is just reduced to Thomson’s then b
_
one resonance
have to compare atoms we have w >wo
we
-
discuss how this
cussion could have been and
--
approximations
If on the other hand
W
< <W 0,
becomes, 2
b
-r,
--
(1-103)
_20 es,.ein-
This is the so-called
Rayleigh scattering, originally proposed to explain the scattering of visible light produced by gasses or small particles. Three important features of this kind of scattering should be noticed: the polarisation factor is the same as for the x-ray Thomson scattering; the scattered amplitude is proportional to the square of the frequency, and the cross-section is thus proportional to the fourth power of the fre-
-
quency;
the sign scattering. -
of the
scattering’ length
is
opposite
to the
one
of the Thomson
point explains the blue color of the sky (the highest frequency which from the first point may appear to be highly polarised. The change of sign noted in the third point is important, since it corresponds to a sign change of (n 1). We shall comment this further when we will dispose from a more quantitative theory. Again for x-rays, the scattering length (1.102), when summed over all the The second
in the visible
spectrum),
-
atomic electrons becomes similar to the and
imaginary
bat where
f
is the Thomson
the correction due to
correction
or
Originally, the
vicinity
it
one
of Thomson
(1.97)
but with real
corrections:
-
in
Y
+
scattering,
resonance.
anomalous was
re
optics
f’
if")6sc- &in,
+
whereas
f’
f",which
3 .
One must take into account
that the anornalous
dispersion
is
real, give. dispersion
are
This correction is the so-called
scattering
of resonances, the
and
(1.104)
as
in
dispersion was introduced. In opposite to the usual behavior for
I
Interaction of
X-rays (and Neutrons)
with Matter
31
the pure Thomson scattering the sum over all the electrons and their spatial distribution, but this discussion is difficult and uncertain in the classical
theory. We shall see that in the quantum theory f’ and f" only slightly depend on q and have an energy dependence that we shall discuss. Though crude, the classical model allows the calculation of the absorption as proposed in the exercise 1.4.1. In fact, one rather gets the total crosssection, including absorption and scattering. This result is very realistic, since it agrees with the prediction of the optical theorem discussed in section 1.3.2. To summarize, the classical model although simple describes most of the phenomena and provides exact values for a certain number of physical quantities. Nevertheless the values of the resonance frequencies and of the damping coefficients are not calculable within this framework and are left arbitrary. In addition, it does not give much indications about the q dependence of the scattering factor at resonance but more important it does not describe the scattering when an electron is ejected (Compton effect). Although it is possible to give a classical description of such an effect by considering the reaction on the scattering of a vibrating electron, only the quantum approach is correct. Therefore the only coherent and completely exact description is given by the quantum theory of the interaction between the radiation and atoms.
1.4.1. Calculate the total cross-section of an atom which exhibits resonance characterised by wo and -/. We assume that the power only taken by an atom from the radiation is the same as the one dissipated by the damping force -/-6 (do not forget that when complex numbers are used to describe the oscillation of real variables, the answer is twice the one obtained Exercise
one
with real
The initial power of the radiation is given in section 1.2.3. optical theorem (section 1.3.2) is satisfied.
numbers).
Check that the
Quantum Description:
1.4.4
a
General
Expression
for
Scattering
and
Absorption
description we shall assume that the radiation is quantised as photons. scattering and absorption probabilities are then the squared modulus of the probability amplitudes. The amplitudes are transformed into scattering lengths and the probabilities’into the scattering cross-section. The amplitudes themselves are derived from a perturbative calculation based on the
In this The
interaction Hamiltonian between the radiation and the electrons. which it is observed that the index of refraction varies in the energy. By extension one refers to "anomalous scattering". adjectives "anormale" and "anomale" are used. "Anormale" not follow the rule and "anomale"
the a
same
species. We
sense as
the
means
that it does
different from other individuals from
dispersion does not constitute only a usual behavior, the second expression seems to be more acknowledge B. Pardo for his comments.
law in itself but
appropriate.
means
same
In French the two
Since the normal behavior of the
Frangois de Bergevin
32
The
expression
contains the
potential
of the Hamiltonian of
following
of the
(1/2rn) (p
-
term
(we
electron in the radiation field
one
leave aside
other terms such
some
as
the
atom)
eA/C)2
P2 /2Tn + (C2 /2 MC2 ) A2
=
_
(e/mc)A.p.
(1.105)
The p and A operators are the momentum of the electron and the vector potential of the radiation. The first term of the right-hand side gives the kinetic energy of the electron and the two others the energy of interaction. In this expression, the spin has been neglected which is permitted when the energy of the radiation is weak
compared to the rest mass energy of the elecperturbation calculation made at the lowest order on the two interaction terms yields the scattered amplitude. This approximation is sufficient because the strength of the interaction, measured by the ratio of the coefficient e2/7,nC2 (or 47reor,) to the quantum size of the electron h/mc (or Ac defined further) is small. The perturbation terms are sketched in Fig. I.A. The smallest order of the perturbation is the first order for the 2 term in A and the second order for the term in A.p. These two terms give rise respectively to one and to two terms in the scattering length (with our convention for the sign of imaginaries, unusual in quantum mechanics): tron which is 511 keV. A
bat
7e
of the
sum
if" arises, corresponds
to
(1.106) an
from which the
excitation energy
dispersion
(or commonly
Frangois de Bergevin
42
a
resonance) E,
-
Ei. The associated correction is
re
(fc
+
ifc’ )
with The real and
x
imaginary parts
I
X
oc X
-
[hw
--
are
-
(E,
-
presented
(1.136)
+
T+X2 Ej)] / (-Vcl2)
1+ X2
2
in
Fig.
1.10.
1.2-
0 8.
0 4-
:3
.
CU
0 0.
f f
-0.4-
-10
-5
0
E
Fig.
1.10. Schematic
nance
at energy
Ec
-
-
5
10
EO/ (r/ 2)
representation of the dispersion correction for a single Eo. f’ and f" are given by b r,(f + f’+ if")
Ej
=
reso-
=
The formulae (1.106) and (1.107) show a correspondence between the dispersion correction in terms of the scattering length and the absorption cross-section. Exactly at the resonance energy, we check the optical theorem (Im is the imaginary part), Cabs
--
2A Im[b
(q
--
0)],
(1.137)
discussed in Sect. 1.3.2. 6 We obtain here
expression for the absorption cross-section, while the optical same expression for the total cross-section. The error comes from our calculation of the scattering length, made in the first order Born approximation. The next order is required to obtain an imaginary part which expresses the intensity loss due to scattering (see appendix LA). The calculation at that order is made intricate because of some difficulties of the quantum theory of ratheorem
an
yields the
(the renormalization of field theory). That error is negligible inasmuch absorption is the largest part of the cross-section, which is true up to moderate energies, but not at the highest. diation
as
the
1
Interaction of
The distribution of the
X-rays (and Neutrons) with Matter
energies E,
resonance
-
Ei, with the edges
-
as
43
main
absorption. Fig. 1.8 shows features, the comparison between the variations of the absorption cross-section close to an edge and the variations of the anomalous scattering. Let us now look how ’the real part of the scattering factor, f + f’, varies when the energy changes from x-rays to near infrared, that is to say from several tenths of keV to one eV. The highest energies are far above the edges of most elements and the Thomson scattering factor f is dominant. For lower energies, a negative contribution f’ appears at every edge and is more important below the edge than above because of the white lines. Low energy edges produce the most intense dispersion effects. Going to low energies, some edges for which f + f is negative are observed, and then a transition occurs towards 10 to 100 eV where f + f’ definitively changes its sign. In this range, the very intense absorption lines enormously reduce the propagation of light in matter, which makes it called vacuum ultraviolet radiation, because it propagates only in vacuum. When the sign of f + f’, which is also the sign of b, changes from positive to negative, the refractive index n goes from below to above the unit value (the link between scattering and the index is discussed has been
in section
1.4.8
previously
discussed about the
1.3).
Resonances:
Dispersion Relations
absorption cross-section is easily obtained directly by experiments as for example, the measurement of the transmission through a known thickness of a material. The imaginary part of the scattering length b is found at the same time. The real part of b however is more difficult to obtain accurately. Among the different methods, diffraction experiments but also reflectivity measurements have been used to extract the scattering length [9]. The drawback of such indirect methods can be overcome because it is possible to rebuild the real part of b if the imaginary part is known over the entire spectral range. Conversely the imaginary part can be deduced from the real one. For a single resonance, if the variation f"(E) as shown in (1.136) and Fig. 1.10 is known, then the energy, the amplitude and the width _V of the resonance are determined, and f(E) can be obtained. Note however that the sign of F is left ambiguous in this procedure and must be given. The question is to know whether such a reconstruction of f’(E) is still possible for the general case with several resonances. It will turn out to be possible, but not in the way it can be done for a single resonance whose shape is known. The key point is not the particular form of the function (1.136) but rather the well defined sign of F. Let us start from the classical model, namely the expression (1.102) for the scattering length,
The
2
b W
2
2
-
WO
_ .
i
7w ’rne ;,.ein-
(1-138)
44
Frangois
de
Bergevin
The
general case can be represented by summing many expressions of this corresponding to each different resonance wo with a different damping constant -y. Since this model is defined by two independent functions of WO, a distribution of the resonance densities and a distribution of the damping constants, one could expect the real and imaginary parts of the scattering length to also constitute two independent functions. However some constraints are imposed because the damping constants -/ are necessarily positive. Although kind
these constraints
seem
to be
weak,
it is remarkable that
to lead to
a
relation between the real and
that such
a
relation does not
come
from
they are sufficient imaginary parts of b. We shall see a particular scattering model; it is
general and concerns the response of any system to an excitation. For the proof, we return to the model with only one resonance (.1-138) but this could be easily extended to the general case. To prove the existence of a relation between the real and imaginary parts of b, it is necessary to make use of a mathematical trick, the analytical continuation of function b in the complex plane. The trick allows one to express some basic properties of complex functions. Indeed b is a complex function of the real variable w. If such a function can be represented by a series expansion which converges for any real value of the variable, then it can also be defined for complex values of this variable. Hence, the series still converges in a domain of the complex plane. Inside this domain, the function that we shall call here O(z) is analytic and follows Cauchy’s theorem. This theorem ensures that for any closed contour C inside the domain of analyticity and for any point z inside the contour, more
OW
I =
27ri
JC
dz’ z
z’ complex,
(z,
Z
C taken in the For this relation to be useful the
integral
positive sense).
must be taken
only
the
region known, plus a curve which continuously approaches infinity in the lower half-plane for instance a semi-circle with a radius approaching infinity (Fig. 1.11, drawn with variable w instead of z). We obtain the wanted relation provided that: (a) the function is analytic in all this half-plane, (b) it approaches zero when the modulus of the variable approaches infinity so that the integral taken over the semi-circle is zero. Under such conditions, relation (1.139) is expressed as an integral over z’ real. These conditions however impose z to be inside the. contour and therefore to have an imaginary part strictly negative though one would wish to have only real quantities. Nevertheless z can be on the real axis, but then the expression on the left-hand side is divided by two since z is at the border (a rigorous proof is available). Finally if P represents the principal part of the integral at the singularity x’ x, then where the function is
over
i.e. the real axis. Let C be the real axis
=
W
Ti
P
j
+00
-.0 -
dx’ X
X
(x, x’ real).
(1.140)
Interaction of
I
X-rays (and Neutrons) with
Matter
45
imaginary parts of this relation can be written separately. This if 0 (x) satisfies the above conditions (a) and (b), i. e. it is analytic in the lower half plane and tends to zero when jxj goes to infinity, some integral relations exist on the real domain of x between its imaginary and real
The real and
shows that
parts.
Imaainary
ci
Real
Fig. 1.11. Integration over a contour C defined by the real axis and a semi-circle having its radius approaching infinity. If a function does not have any pole inside the contour it satisfies relation (1,139) (w has the same role as the z variable). The poles of the scattering length b(w) (1.138) have been represented. They are outside the contour
The
written in
scattering length
(1.138),
which is
a
polynomial
fraction
of the variable w, is analytic over any domain which does not contain its poles, i.e. the zeros of its denominator. These zeros, indicated in Fig. 1. 11, are i7/2 w, P, depends on wo and -y). The scattering length b satisfies condition
(a)
because the
which b
by
are
w.
constant -y which is
damping
in the upper half
plane.
necessarily positive yields poles
satisfy
To
but it is better to divide
going addany pole
we are
by W2
to comment. Let
and does not
us
since
we
(b) one could divide b(w)lw (after replacing O(x)),
condition
The relations would then be valid in then
get
more
general
notice that the division of b
change
the domain of
by
relations w
2
as
does not
analyticity.
(1.140) is not yet completely convenient because the domain does not extend over the entire real axis but only over its,
A relation such
physical positive side (the oo
as
variable is the radiation
frequency). Integrating
from 0 to
is however sufficient since b verifies
b(-w) We should look if such
a
--
b*
(w).
(1.141)
symmetry of the scattering length is attached
to
A Fourier transform which transforms
particular model, or more general. expression from w to time space shows that this is simply the expression of a symmetry by time reversal. It is thus a general property which a
the above
Frangois
46
de
Bergevin
has however
a limitation: this symmetry does not hold for magnetic moments following expressions do not holdfior magnetic scattering. In that case, the equality (1.141) is written with a minus sign and different relations are obtained.’ With this equality ,tnd. a bit of algebra, one can rewrite the real and imaginary parts of (I. 1 0). Replacing 0 by b (w) lw’ and x, x’ by w, W’ yields so
the
Re [b (w) 1w
2
2]
2wp
_Emfb (W) /W 2] These relations
are
w’lm[b (wl) /U) /21dw’
P
7r
7r
W
0
JO
/2
_W
2
Re[b(W’)1W12] d.,.
’
W12
-
(1.143)
W2
the so-called Kramers and
Kronig
(1.142)
or
dispersion
relations for the
scattering length. model, the proof we have given
poles of b(W )/W 2 positive value of the damping constant but it can also be inferred from the principle of causality, which is of very general extent. To understand the equivalence of these two hypotheses, positive value of the damping constant and principle of causality, it is worth returning to the resolution of the differential equation (1.99), In this
assumes
all the
to be above the real axis. We have inferred this from the
which describes the movement of the electron in the incident field. We rewrite this
the
equation by noting z in
The
u
properties of
displacement
radiated field which is
proportional
mii +
A
u
instead of z to avoid any confusion
the present section; for simplifying, u will be a scalar. that we are going to discuss now are also the ones of the
with the variable
-lit
to
u.
2
+ mw,u
-
(-e)Eo e
iwt
(1.144)
systematic method to solve such a differential equation with a right-hand f (t) consists in using the Green function of the equation. This method
side
has been described in section 1.2.5. Let of
equation
is
us
recall that
a
solution of this kind
given by +00
U(t)
UO(t)
G (t
+
-
t) f (t’)
df
+00
G(t
-
t’)f (t’) dtl,
(1.145)
-00
function, G(t), is solution of the equation with J(t) instead f (t) right hand side; the solution uo (t) of the homogeneous equation the right hand side) becomes nearly zero after a certain amount of (without time due to damping. Writing the electron displacement u(t) as in (1.145)
where the Green of
7
in the
In practice the same dispersion relations can be written for magnetic and non magnetic scattering lengths provided that the magnetic part is affected by a factor i.
I
Interaction of
X-rays (and Neutrons) with Matter
47
to be
given. The displacement u of the excitation superposition given action f at any time t’; since the laws are invariant by time translation, the t’. G(t) is obtained through coefficient G only depends on the difference t its Fourier transform g (w). We replace the right-hand side of equation (L 144) by (t), whose Fourier transform is one. The derivatives in the left-hand side transform into powers of w, so we get allows the at
a
following physical interpretation
time t is the result of the linear
-
g
which
(W)=
21
1 -
MW
2
_W0_
(1.146)
i7w1rn)
yields G(t) +00
G (t)
27rm
eiwt W2
-
W20
-
i7W/M
(1.147)
dw.
integral, it is possible to integrate along a closed path in complex plane: if the function does not have any pole inside the path of integration, its integral over it is zero. The poles of g(w) are those of the scattering length that we have just discussed; the integral taken over the path of integration C (Fig. 1.11) is then zero. For t < 0 the integral over half the circle is also zero since the numerator is bound and the integral of 0 for t < 0. dw / I w12 goes to zero when I w I goes to infinity. Then G(t) It is important to mention that the proof depends on the position of the poles of g(w), and on the positive sign of the damping constant. Conversely if G(t)-- 0 for t < 0, it can be shown that g(w) does not have any pole below the real axis and the Kramers-Kronig relations can be applied to g(w) and u(W). The condition G(t) equal to zero at negative times constitutes the expression of a causality principle, according to which an excitation given at a certain Z.nstant cannot produce any effect before this instant. Making the dispersion relations to depend on this principle gives them a very general extent, beyond the cases where it is possible to clearly define some damping. In our world most of the phenomena are irreversible and time is therefore asymmetric. The positive character of the damping and the principle of causality as discussed here, both constitute two manifestations of the irreversibility. We have proved that one or the other of these two principles yields the dispersion relations (1.143). But a question still remains. It is generally admitted that microscopic laws in physics are mainly symmetric with respect to time reversal. On the opposite, the irreversibility manifests itself in macroscopic phenomena and in-statistical thermodynamics. We may be surprised that irreversibility is invoked in the scattering of radiation by an atom, which seems to be a rather elementary microscopic phenomenon. However one can also see that scattering involves some disorder. A plane wave travelling in vacuum, such as the incident wave constitutes a very unlikely state that can be considered as out of equilibrium. The ground state of a radiation is made of random waves in thermal equilibrium with neighbouring objects. Even considering a monochromatic radiation inside a perfectly reflecting box, it is only To calculate this
the
--
Frangois de Bergevin
48
possible
to find the initial
wavelength that has been scattered into multiple plane waves. One can see that the scattering of a plane wave by an atom is irreversible, a bit like the dilution of an alcohol droplet in a glass of water. The final state is the spherical wave moving away from the atom, superimposed to the incident wave which has a reduced amplitude. Therefore the unique incident plane wave has been changed into a superposition of plane waves travelling in all the directions. In addition, any of the plane components of the diverging wave can be associated to a particular movement of disordered
the atom since the momentum must be conserved. This is reminiscent of the dilution effect. If the
scattering were reversible, one could produce the reverse operation: starting from a spherical wave converging towards an atom and from a plane wave, one could see the plane wave coming out with an increased amplitude. This would be difficult to realise and may be impossible. For this, one should correlate the different plane components of the converging wave to some particular movements of the atom. The difficulty is similar to the one
hol
which would be faced in
droplet by imposing
initial conditions such
an
attempt
to invert the dilution of the alco-
to the molecules of the water-alcohol mixture
the mixture would demix into two
some
a phases the simplified problem not taking the absorption and the fluorescence in account. Multiple photons are then re-emitted for only one absorbed; in that case the radiation as
few instants. To what must be added the fact that
becomes still
more
after
have
we
disordered and this event is less reversible. As
a
matter
of fact it appears that absorption and resonant scattering contribute much more to the dispersion than pure elastic scattering. From the above arguments one can be convinced that even though the scattering looks like an
elementary phenomenon, it Zs actually something irreversible, which has obey the dispersion relations associated with irreversibility.
1.5
1.5.1
to
X-Rays: Anisotropic Scattering Introduction
we present briefly some other types of x-ray scattering, obessentially in crystalline materials. These are the magnetic scattering, which depends on the magnetic moment of the atom, and the Templeton anisotropic scattering, which depends on the neighbourhood of the atom in the crystal. A common feature to these two scattering effects is their anisotropy. The usual scattering amplitude which is described in the previous sections can be said isotropic because it depends on the incident and scattered polarisation directions through a unique factor, s,.Zi, independent of the orientation of the scattering object. The atomic scattering amplitudes which we discuss now can be said anisotropic because they depend on the
In this section
served
orientation of the characteristic
dent and scattered
polarisations.
axes
of the atom with
The characteristic
respect to the incimay represent the
axes
1.
Interaction of
X-rays (and Neutrons) with Matter
magnetic moment direction of the atom if it exists, or the directions crystal field which eventually perturbs the state of that atom. These
scattering
nisms. The first
effects
can
take their
origin
’49
of the
from two different mecha-
is the interaction between the
electromagnetic radiation spin of the electron. It produces some scattering, the so-called non resonant magnetic scattering. This one is essentially independent of the binding of the electron in the atom, as is the Thomson scattering. A second type of anisotropic scattering arises as a part of the anomalous or resonant scattering, presented earlier in section 1.4.7. The atomic states I i > and one
and the
I
>
c
in
(1.106),
that is the initial state and the
one
in which the electron is
anisotropic. If that anisotropy originates from a magnetic moment, the resulting scattering is called resonant magnetic scattering. If the anisotropy is some asphericity of the atom kept oriented by the peculiar symmetry of the material, it is the Templeton anisotropic scattering.
promoted
may be
X-ray magnetic scattering is It
can
be used with
some
a
useful
complement
elements whose
common
to neutron
scattering.
isotopes strongly
absorb
thermal neutrons. The very good resolution (in all respects, position, angle and wavelength) of x-ray beams is an advantage for some studies. Since xray
scattering depends
on
some
characters of the
magnetic
moment in
a
way different from neutrons, it may raise some ambiguities left by neutron scattering experiments. One of these ambiguities is the ratio of the orbital
spin moment of the atom, because they contribute to neutron scattering exactly in the same way and. cannot be discriminated from each other. xray amplitudes given by spin and orbital moment depend differently on the geometry of the experiment and they can be separated out. The resonant xray magnetic scattering is element dependent and eventually site dependent, which may give some useful information. It is also a spectroscopic method which probes the electronic state of the atom. The availability of small and brilliant x-ray beams compensates for the smallness of magnetic amplitudes in the study of thin films and multilayers. When the magnetic element has a very intense resonant magnetic scattering, even a single atomic layer can be probed. to
Applications of Templeton anisotropic scattering are not yet fully developed. It can give some information on the orbital state of the atom in a crystal. Quite recently, a strong interest has developed about the so-called orbital ordering, where an alternating orientation of atomic orbitals can be detected with the help of this type of scattering. In the present section we give a short description of the non resonant magnetic scattering, the resonant magnetic scattering and the Templeton anisotropic scattering. We also discuss the anisotropy of the optical index. The case of the magnetic neutron scattering is described in Chap. 5 of this
book.
Franigois de Bergevin
50
-e
Reradiation
Force
Driving
E
*
E
E
E-dipol.
H
/T\ it \-/
E
H-quadr. E
grad
H pt
E-dipol. H
H
/T\
Torque HxjL
H-dipol. H
-e
Hxp/m E-dipol.
Fig.
1.12. The electron
can
scatter the
electromagnetic
radiation
through
a
variety spin pair
of processes. In each of them, the incident field moves the electron itself or its through a driving force on the left. The back and forth motion is indicated by a
of thin
opposite
arrows.
In this
motion, the electron re-radiates through
a
mode
The first process is the well known Thomson scattering. the scattering by the spin, drawn as a double arrow. The describe 2 4 to Processes line is the fifth in a correction to Thomson scattering when the electron process
indicated
has
a
on
the
translation
right.
motion, indicated by the momentum p. When integrated gives rise to a scattering by the orbital
orbit of the electron in the atom, it
over
the
moment
Interaction of
I
1.5.2
can
51
Magnetic Scattering
Non Resonant
Similarly
X-rays (and Neutrons) with Matter
to the Thomson
the
scattering,
be found either in the classical
resonant
magnetic scattering
quantum theory. The quantum calcu-
or
[10].
non
spin of the electron is associated a a magnetic description, interacts with the which, radiation. The the of Fig. 1.12 shows schematically how magnetic component the interaction between the electromagnetic field and the electron, comprised of an electric charge and a magnetic moment can produce a magnetic dependent scattering. Having some interplay between spin and motion in space, and having some magnetic properties attached to an electrically charged particle lation
be found in reference
can
with
are
The
classical
in
moment
relativistic effects. That relativistic character introduces the scale factor
jhqj /27rme between the
magnetic
and Thomson
(1.148)
(A,/A) sin 0,
2
--
scattering amplitudes of an electron. In a typically of the order of 10-2 Since
that scale factor is
diffraction
experiment only unpaired electrons, which are at most one or two tenths of all electrons of a magnetised atom, contribute to the magnetic scattering, the magnetic amplitude is in favorable cases 10-3 10-4 of the Thomson amplitude. The intensity of magnetic Bragg peaks of antiferromagnets is then affected by a factor of the order of 10-7 The orbital moment contributes to the elastic .
_
.
spin moment and with the same order of magnitude, but with a different dependence on wave vectors and polarisations. We write below the scattering length of an electron of spin S and orbital moment L
scattering
as
bmag
well
ir,
-
The tensors
as
the
(A,/A)
[(ee*
sc
Ti S . &j.
Ts, TL simply help
polarisations. angle between
Their elements
iin
and
(8) (P)
(
are
sc
T:L
.’ jj) L] .
.
(1.149)
to write these bilinear functions of the
vectors. In their
expression below,
20 is the
sc: (P)
(8) TS
S+
isc X iin 2isc sin 2 0 -2iij sin 20 sc X iin
(1.150)
(P) TL
(8) (P)
(
(is, iin)
0 -
( s iin) c
+
+
sin
2
0
2isc
x
sin 20
(1.151)
iii sin 20
Remember that we use for i a sign opposite to the one used in quantum theory in the frame of which these equations are usually written. Magnetic Compton scattering is also present but results only from the spin. Since hq can be an important fraction of mc, the magnetic Compton amplitude can be significantly larger than the elastic one.
Frangois de Bergevin
52
Resonant
1.5.3
Magnetic Scattering
explained in sections 1.4.4 the resonant, or dispersive, part of the scattering is based on the virtual excitation oi. an electron from a core level to an empty state, which can be just above the Fermi level. In the subsequent discussion in section 1.4.7, we have assumed that the polarisation factor was This assumption in fact may the same as for Thomson scattering, be wrong. Let us write the numerator of a particular term in bdispl (1.106), while making the dipolar approximation (the exponentials are reduced to 1) As
ic,
a3
being nine real coefficients, this is the most general tensor expression (1.152). The actual structure of that tensor is determined by the symmetry of the scattering atom. The spherical symmetry is frequently a good approximation, though never completely exact in a With ai, bi, representing
the
crystal;
then
T,,, reduces
factor
sc *-ein
A
case
of
ci
and
we recover
the usual
of the symmetry is the presence of a magnetic moment. time inversion, which should not change the scattering
a
amplitude, exchanges the
magnetisation.
incident and
* 0 4C
reverses
-the
T,,,, that is the
the form
(1.154)
X-
magnetisation, at least in the simplest cases. arguments should be completed by an explicit discussion
the direction of
These symmetry
physical process. shortly explained in Fig.
of the
3SC Tres -’ in
i
and
of
antisymmetrical part magnetisation. That part has cc
just
scattering beams
This shows that the
is of odd order in the
where i is
[11]
-
lowering
We may observe that
ici’s,
to the unit matrix
0C
The mechanism is described in reference 1.13 and
&SC -’ in Vi I
F1
+ +
caption. -
1)
+ i
(" *SC
X
iin)
( isc -2) (’ in -i) (2F, o
-2 -
Vi 1 F, I
-
-
F1
F,
-
-
[12]
and
crystal field,
In the absence of any
1)
1)
,
(1.155)
where FI-1, Flo and F11 contain some transition probabilities. These transitions are described by two indices, the first one standing for the change in the orbital moment AL
(I
in the
dipolar term),
and the second
one
for
Interaction of X-rays
I
(and Neutrons)
with Matter
53
Energy down Fe rmi
Lui
2P3/2
magnetic scattering in the case of the Lrlr as platinum. Due to the magnetic moment the resonance occurs preferentially in the spin down (S -1/2) half valence band, on the right. Because of the strong ’Spin-orbit coupling in the core shell 2p, the 2P3/2 level is completely separated out from the 2p,/2 and contributes
Fig.
1.13. Mechanism of the resonant
resonance
of
a
third
row
transition element, such
=
alone to the is
resonance.
netisation)
Therefore the
(S
=
-1/2)
state involved in the
resonance
rather defined value of the z- component (:F the direction of magof the orbital moment in the initial state of the electron. For a given
coupled with
a
polarisation of the radiation, this makes the amplitude magnetisation direction, here the up direction
to
depend
on
the atom
isotropic anomalous scattering, discussed in section just discussed antisymmetrical part. The third one depends on the axis along which the magnetisation is lying, but not on its.sign; it, is responsible for the magnetic linear dichroism. One should not forget that the above expression is to be multiplied by a resonance function of the energy showed in (1.136) and Fig. 1.10. AL, The first
term is the
1.4.7. The second term is the
The spin orbit coupling is a key feature of this mechanism. In the example displayed in Fig. 1. 13, the spin orbit coupling interaction is very large in the core
state, but in
some cases
only in the excited state. In formula, a quadrupolar one neglected if it corresponds to a tran-
it may be present
term written in the above
dipolar Though smaller, it cannot be sition to a strongly magnetised atomic shell such as the 3d shell of transition element ’, or 4f shell of lanthanides. It shows a quadrilinear dependence on i SC iin, k,;,, iin addition to the also exists.
i
-
The order of
magnitude
of the resonant
magnetic scattering
may vary
on
wide range. The K resonances, accessible for example in the 3d transition elements, have amplitudes which are comparable to the non resonant. Indeed a
the K shell has
no
orbital moment
so
that the effect relies
on
the
spin
or-
54
Frangois
’
de
Bergevin
bit
coupling in the valence shell, which is much less efficient. Furthermore dipolar transition occurs to a weakly magnetised p valence shell, while the strongly magnetised d shell caD give only a quadrupolar transition. The latter drawback limits also t1le L.jjjjr resonances of lanthanides, but then the L shell is completely split by the spin orbit interaction. In that case the amplitude is typically ten times larger than the non resonant. The Ljl-,.rjj the
of transition elements
resonances
are
favorable in all respects and enhance
several orders of
amplitude by magnitude compared to non resonant. In the of 3d case elements, long wavelength (of the order of 1.5 nm) can only fit long periodicities, and give diffraction on multilayer, or be used in reflectivity experiments. The L resonances of 5d transition elements arise in the 0. 1 nm range but among those only the platinum group elements, and mainly the platinum itself, can take a magnetic moment. The Miv,v resonances of actinides offer the same favorable characters and are also quite effective, with amplitude enhancement by a factor of the order of one thousand. The wavelength, near 0.35 nm for the uranium allows for Bragg diffraction experthe the
iments.
Templeton Anisotropic Scattering
1.5.4
Even without any magnetic moment, the atom may show a low symmetry. Most often, this arises from the crystal field. A spontaneous orbital order
(that
arrangement resulting mainly from the electrostatic inneighbouring atoms) is also expected in some materials. As a consequence the symmetrical part of (1.153), aj, bi, differs from the unit matrix. Again a quadrupolar term may exist. The Templeton scattering produces some change in the intensity of Bragg peaks at the absorption edges and this can be used to get more structural information. A striking feature is the occurrence of otherwise forbidden reflections [13- 5]. is
an
orbital
teraction between orbitals of
When
reflection is forbidden because of
a
scalar
(that
axis
a screw
or
glide plane,
the
independent on axis, the atom rotates from one site to the next, so that the tensor amplitude in (1.152, 1.153) may not cancel. This breakdown of a crystallographic extinction rule should not be confused amplitude
of the
cancels
atom).
For
if it is
only example
a
with
is
the orientation
a screw
with the appearance of e.g. the 2 2 2 reflection in the diamond structure. In that case the structure factor is a scalar and the broken extinction rule is not a
general
rule for the space group; it
applies only
to
a
special position
in the
cell.
1.5.5
The Effect of
If the
is
tical
The
scattering properties.
an
Anisotropy in
anisotropic, non
so
the Index of Refraction
may be the index of refraction and the op-
resonant
magnetic scattering amplitude is
zero
in
the forward direction. From the discussion in section 1.3 it cannot contribute to the refractive index. We shall therefore discuss
only the
consequences of
Interaction of
I
X-rays (and Neutrons) with Matter k
55
Templeton scattering. We give only some brief inquestion which could deserve quite a long development. The propagation of neutrons in magnetised materials and the associated reflectivity is examined in the chapter 5 of this book. It is different from, and somewhat simpler than the propagation of the electromagnetic radiation in an anisotropic medium, especially when the interaction is strong. The thermal neutron has a non relativistic motion which allows for a complete separation of the space and spin variables. The direction of propagation is in particular independent of the spin state. The electromagnetic radiation instead is fully relativistic, which intermixes the propagation and polarisation properties. In that case, the direction of propagation depends on the polarisation. Several unusual effects are consequently observed. For example the direction of a light ray may differ from the normal to the wave planes, or a refracted ray may lie outside of the plane of incidence. resonant
magnetic
formation
and of
this
on
Starting from the Helmholtz equation (I 15) -
modifies the dielectric constant we
wrote the Helmholtz
be of fourth tensor we
order,
-_
n’)
of the medium
anisotropy
which becomes
a
tensor.
Though
for the 4-vector A and the tensor should
all the useful coefficients
are
contained in
in space, similar to (1.153). of crystal optics, described in several
a
third order
In the absence of
acting
have the
(c/,Eo
equation
the
case
magnetisation, textbooks, e.g. [3]. If
magnetised, the antisymmetrical part of the tensor is non zero and some phenomena occur, such as the Faraday rotation of the polarisation plane or,,the magnetooptic Kerr effect (that is polarisation dependent and polarisation rotating reflectivity). The basic theory can be found in [16]. The theory as exposed in the textbooks is drawn in the dipolar approximation, which is legitimate in the range of the visible or near visible optics. I am not aware of a complete description of anisotropic optics including the quadrupolar terms. It seems reasonable in practice to use instead of such a full theory, some perturbative corrections since the quadrupolar resonance terms are always small. The incidence of the quadrupolar term is clear in an effect well observed with visible light and discovered nearly two centuries ago: it is the optical activity, that is the rotation of the polarisation plane in substances the medium is
which lack
a
center of
symmetry. Indeed the combination of
a
effect. It is
dipolar
and
a
small
effect, produce quadrupolar terms is required since it corresponds to differences of the order of 10-4 between the indices of the two opposite circular polarisations. Yet it can be easily observed because the absorption of the visible light is still smaller and samples more than 10’ wavelengths thick can be probed. to
such
an
a
In the x-ray range, at energies of several keV and above, the refractive index differs from one by a small value and its anisotropic part is still smaller.
Only a limited ple terms. One dichroism. In
a
list of effects
are
observed and
they
are
interpreted magnetic
of the most studied of those effects is the
or ferrimagnet, where a net magnetisation is present, changes, according to the helicity of circularly polarised
ferro
the refractive index
in simcircular
Frangois de Bergevin
56
being parallel
antiparallel
to the
magnetisation. Indeed the optical (1.69) (1.83) yields the absorption atomic cross section or the dispersive part of the index of the medium, from the part of the scattering length written in (1.152, 1.155). For that we make ,,, equal to in and equal toi_ in. For a circular polarisation, the term (1.154) is real and x-rays
theorem
or
or
its extension
reads
+kin.i. The
for the
right handed helicity to + for the left complex resonance factor which we have left out from the formula. The difference in the real, S, component of the index, between both helicities gives rise to the Faraday rotation of the polarisation plane. Similarly the difference in the imaginary, , component gives rise to a difference in the absorption, called the magnetic circular dichroism. Once a circularly polarised radiation is available, it is relatively easy to measure that change of absorption, usually by switching the magnetisation parallel or antiparallel to the beam. Similarly to the resonant scattering, the dichroism shows a spectrum in the region of the absorption edge. At the L edges of the 3d elements, the resonances and their magnetic parts are quite large and the full optical theory, in the dipolar approximation, should be considered. Some reffectivity measurements have been done, e.g. sign
is switched from
(1.156)
handed. This is to be
-
multiplied by
the
[17]. LA
Appendix:
Anne
Sentenac, Fran ois Vignaud
Approximation
the Born de
Bergevin,
Daillant,
Jean
Alain
Gibaud,
and Guil-
laume
appendix we give the Born development for the field object. In absence of any object, the field (scalar for simplicity) is homogeneous Helmholtz equation,
In this
scattered
by
a
deterministic
( A + k2) Ain (r) 0
The
object introduces
(1.16-17).
In this
case
a
perturbation
V
on
the field is solution 2
(ZA + ko
-
--
solution of the
(LAI)
0.
the differential operator,
see
Eqs.
of,
V(r)) A(r)
--
0.
(LA2)
sum of an incident field Ain (r) homogeneous equation) and a scattered field A, which satisfies the out-going wave boundary condition. Following section 1.2.5, we transform Eq. (LA2) into an integral equation by
The total field A
can
Aine- iki,,.r
wave
(plane
be written
as
the
solution of the
=
(and Neutrons)
X-ray,
G_
outgoing
that satisfies
wave
(LA3)
-
47r
+J
power of the convolution
A(O)(r)
-ikor
I --
r
G
-
(r
AM
--
We obtain
r) V (r’) A (r) dr’.
-
write the solution of this
one can
A
with,
(r)
boundary condition.
Ai,, (r)
A
series in
57
the Green function
introducing
Formally,
with Matter
(LA4)
integral equation
in terms of
a
operator [G-V], +
AM SC
(2) + A SC +
(LA5)
Aijr), A(’) (r)
d 3r’G
SC
Jd r’J
A(2)
3
SC
d 3r"G_
(r
(r.
-
-
r’) V (r’) Ail (r%
-
r’)V(r)G- (r’
-
r")V(r")Ai1, (r").
When this series is convergent, one gets the exact value of the field. The main issue of such an expansion lies in its radius of convergence which is not easy
Physically,
to determine.
potential V(r)
the
combined with the
propagation
operator G_ represents the action of the particule (or polarisation density) at r/ on the incident wave, i. e. a scattering event. When the potential appears (in the first order term) the incident wave is singly scattered by the once,
object. When it appears twice (in the second order term), scattering events, etc. Eq. (LA5) can also be viewed as a perturbative development in which the scattering event [GV] is taken as a small parameter. The first Born approximation consists in stopping the development in Eq. (LA5) to the first order in V (thus assuming the of the
particules one
accounts for the double
predominance of single scattering). We now proceed by evaluating the scattered far-field and the scattering crosssection. We assume that the observation point r is far from all the points r’ constituting the object (with respect to an arbitrary origin situated inside the
object).
In this case,
one
has
Ir 8
In operator notation
1/1
_
which
X
yields A
with
+X+X2 +
=I
==
....
can
rl
make
Indeed,
,:z
r
an
-
(LA6)
ia.r’,
analogy with the Taylor expansion
the field
can
be written
as
A
=
Ai./i
the series
Ail +
[G-V]f
one
-
=
[G-V]Ain
f G-(r
-
+
[G-V][G-V]Ain
r’)V(r’)f (r)dr.
+... +
[G- V]n Ai. +...,
-
of
[G-VI
Frangois
58
so
de
Bergevin
that
G_
(r
-
e-ikor
r’)
Ir
47r
r’l
-
Using this far-field approximation given in section 1.2.4,
in
A,,,(r)
4r
Eq. (LA4),
-Ainb(i i)
eikoia.r’.
(LA7)
r
one
retrieves the
expression
(LA8)
r
Bearing in mind the Born development for the field, one can write the scattering length b in the form, b(fi) W)(ii) + b (2) (ji) + with, for example ...,
I
b
d 3,r/ V(r’) ei(kofi-ki.).rl
(LA9)
.
47r
The calculation of the differential
scattering cross-section, Eq. (1.36),
do-
dQ
lb(fi) 12
is then
straightforward. noting that the perturbative development of the energy (which is proportional to the square of the field) starts at second order in V. Hence, to be consistent in our calculation, we should always develop the field up to the second order to account for all the possible terms (of order two) in the energy. A striking illustration of this remark is that the first Born approximation does not satisfy energy conservation. This can be readily shown by injecting the perturbative development of b in the optical theorem which is a direct consequence of the energy conservation. The optical theorem relates the total cross-section to the imaginary part of the forward scattered amplitude. One has, see section 1.3.2, o-t,,t 2A_Tm[b(ki,1)]. If one disregards lossy media, the total cross-section is equal to the scattering cross-section, It is worth
=
O’S.c
-_
J I (iii) 12 b
dQ
=
2A_Tm[b(ki,1)].
(LA10)
expansion Eq. (LA5) being a formally exact representation of the field, optical theorem, written as a series, is verified at each order of the perturbative development. One gets, to the lowest order, The
the
II
b (1) (fi)
12 dQ
--
2A1m[bM(ki,,)
This last
equation shows clearly that if one
conserve
energy,
order in the
one
forward
+ b (2)
wants the Born
should calculate the scattered direction.
(ki,,)].
(LAII)
approximation to amplitude up to second
Interaction of
I
X-rays (and Neutrons)
with Matter
59
References 1.
Landau, E.M. Lifshitz, Course of theoretical physics vol. 8, Electrodynamics media, Pergamon Press, Oxford, 1960. L. Landau & E. Lifchitz. Physique th6orique. Tome VIII, Electrodynamique des milieux continus. Ed. Mir, Moscou (1969). J. D. Jackson. Classical Electrodynamics. 2nd Ed.. John Wiley & Sons (1975). M. Born & E. Wolf. Principles of Optics. 4th Ed.. Oxford, New-York (1980). R. W. James. The optical principles of the diffraction of X-rays. The cristalline state, vol 11, Sir Lawrence Bragg ed.. G. Bell & Sons ltd, London (1962). W. Schfilke. Inelastic scattering by electronic excitations. In Handbook of synchrotron radiation, vol. 3, G. Brown & D. E. Moncton ed.. Elsevier Science
L.
of continuous
2. 3.
4.
5.
Publisher 6.
7.
(1991).
Berestetskii, E.M. Lifshithz, L.P. Pitaevskii, Course of theoretical physics vol. 4, Quantum electrodynamics, Pergamon Press, Oxford, 1982. E. Lifchitz & L. Pitayevski (L. Landau & E. Lifchitz). Physique th6orique. Tome IV, Th6orie quantique relativiste, Premi6re partie. Ed. Mir, Moscou (1972). International tables for x-ray crystallography; vol. III Physical and chemical tables; vol. IV Revised and supplementary tables to vol. 11 and 111, The Kynoch
V.B.
Press, Birmingham, 8.
elements Z 9. F.
=
I to Z
Stanglmeier,
48, 626 10.
1968.
E. Storm & H. 1. Israel. Photon B.
=
cross
sections from I keV to 100 MeV for
A7, 565-681, (1970). G6bel, M. Schuster, Acta Cryst.
100. Nuclear Data Tables
Lengeler,
W.
Weber,
H.
A
(1992).
Blume, J. Appl. Phys, 57, 3615-3618, (1985). Templeton and L.K.Templeton, Acta Cryst. A 36, 237 (1980). J. P. Hannon, G. T. Trammel, M. Blume & Doon Gibbs, Phys. Rev. Lett. 61,
M.
11. D.H. 12.
15.
1245-1248, (1988). Dmitrienko, Acta Cryst. A 39, 29-35, (1983). D.H. Templeton and L.K. Templeton, Acta Cryst. A 42, A. Kirfel and A. Petcov, Acta Cryst. A 47, 180 (1991).
16.
A. V. Sokolov.
17.
C. C.
13. V. M. 14.
478
(1986).
Optical properties of metals. Blackie and Son Ldt, London
(1967). G.
Kao, C. T. Chen, E. D. Johnson, J. B. Hastings, H. J. Lin, G. H. Ho, Meigs, J.-M. Brot, S. L. Hulbert, Y. U. ldzerda & C. Vettier, Phys Rev. B
50, 9599-9602,
(1994).
Statistical
2 at
Aspects
of Wave
Scattering
Rough Surfaces
Anne Sentenael and Jean Daillant2 ’
2
LOSCM/ENSPM,
Universit6 de St Hr6me, 13397 Marseille Cedex 20, France, Physique de 1’Etat Condens6, Orme des Merisiers, CEA Saclay, 91191 Gif sur Yvette Cedex, France Service de
Introduction
2.1
The surface state of
objects
in any scattering experiment is, of necessity, of the most varied nature and lengthscales, ranging
rough. Irregularities from the atomic scale, where they are caused by the inner structure of the material, to the mesoscopic and macroscopic scale where they can be related are
to the defects in
processing in the case of solid bodies or to fluctuations in liquid surfaces (ocean waves, for example) The problem of wave scattering at rough surfaces has thus been a subject of study in many research areas, such as medical ultrasonic, radar imaging, optics or solid-state physics [1], [2], [3], [4]. The main differences stem from the nature of the wavefield and the wavelength of the incident radiation (which determines the scales of roughness that have to be accounted for in the models). When tackling the issue of modelling a scattering experiment, the first difficulty is to describe the geometrical aspect of the surface. In this chapter, we are interested solely by surface states that are not well controlled so that the precise defining equation of the surface, z z(x, y) is unknown or of little interest. One has (or needs) only information on certain statistical properties of the surface, such as the height repartition or height to height correlations. In this probabilistic approach, the shape of the rough surface is described by a random function of space coordinates (and possibly time as well). The wave scattering problem is then viewed as a statistical problem consisting in finding the statistical characteristics of the scattered field (such the
case
of
=
as
the
mean
the surface
value
or
field correlation
functions),
the statistical
properties of
being given.
In the first section of this contribution
we present the statistical techniques rough surfaces. The second section is devoted to the description of a surface scattering experiment from a conceptual point of view. In the third section, we investigate to what extent the knowledge of
used to characterise
the field statistics such
as
the
mean
field
for
or
field autocorrelation is relevant
interpreting the data of a scattering experiment which deals necessarily with deterministic rough samples. Finally, we derive in the fourth section a simple expression of the scattered field and scattered intensity from random rough surfaces under the Born approximation.
J. Daillant and A. Gibaud: LNPm 58, pp. 60 - 86, 1999 © Springer-Verlag Berlin Heidelberg 1999
Statistical
2
Description
2.2
2.2.1
Aspects
of Wave
Scattering
at
Rough
Surfaces
61
Randomly Rough Surfaces
of
Introduction
example of a liquid surface. The exact morphology rapidly fluctuating with time and is not accessible inasmuch as the detector will integrate over many different surface shapes. However statistical information can be obtained and it provides an useful insight on the physical processes. Indeed, these fluctuations obey Boltzmann statistics and are characterised by a small number of relevant parameters such as the density of the liquid or its surface tension (see Chap. 9). Let
us
first consider the
of the surface is
origin (such as metallic treatments (like similar technological undergone ) optical all the microscopic to it is Since and reproduce impossible cleaning). polishing factors affecting the surface state, these surfaces have complex and completely different defining equations z z(x, y). However, if the surface processing is well enough controlled, they will present some similarities, of statistical nature, that will distinguish them from surfaces that have received a totally We
now
consider
set of surfaces of artificial
a
that have
mirrors
--
different treatment.
examples,
In these two
we
are
faced with the issue of
describing
a
set of
real surfaces which present similar statistical properties and whose defining equations z (x, y) are unknown or of small interest (see Fig. 2.2. 1). It appears convenient [2] to approximate this set ’of surfaces by a statistical ensemble of surfaces that
are
realisations of
a
coordinates r1l (x, y), parameters of the physical processes size of the polishing abrasive in the
likely that
random continuous process of the plane properties depend on some relevant
whose statistical
-_
affecting case
the characteristic functions
the surface state
(like the grain origin). It is
of surfaces of artificial
z(rll)
of the surfaces
generated by
random process will be different from that of the real surfaces under but the statistical properties of both ensembles should be the same.
2.2.2
the
study,
Height Probability Distributions
Generally speaking, a random rough surface is completely described statistically by the assignement of the n-point (n -+ oo) height probability distribution p,, (r,11, zi r,, 11, z,,) where p, (r 111, zi r.11, z,,)dzi dz.,, is the probability the height coordinates for the surface points of plane rjjj,...r,jj of being at ...
...
...
dzi z,, + dz,,). However, in most cases, we restrict the description of the randomly rough surface to the assignement of the Inone and two-points distribution functions pi(r1j, z), and P2(rill, zi; r211, Z2)information. this need theories solely deed, most scattering From these probability functions, one can calculate the ensemble aver-
between
(zi
...
z,,)
and
(zi
+
...
age of any functional of the random variables
(zi
...
z,)
where zi
=
z(rjjj,,,,),
Anne Sentenac and Jean Daillant
62
7.0
-A_o
_=o
20-0
Fig.
Examples properties
of various
2. 1.
tistical
through
the
2-
rough surfaces
that present the
same
Gaussian sta-
integral, oo
(F)(rjjj r,,Il)
f
--
...
F (zi
...
z,,)p,, (r 111, z,
...
r,,Il,
zn)dzj
...
dz,,.
(2.1)
oo
The domain of integration covers all the possible values for (z, Zn). This quantity is equivalent to an average of F calculated over an ensemble of ...
surface realisations
Sp, N
(F)(rjjj r,,Il) ...
F(zpl..’.zPn),
(2.2)
surface realisation at
plane coordinates
lim
=
n
N-+oo
P=1
where
zjP
is the altitude of the
p-th
rjjj. With this
definition,
one
obtains in
particular the
mean
height
of the surface
through
(z) (r I I) The
mean
square
height
z
(rll)pl (r1j, z) dz.
of the surface is
(2.3)
given by
oo
(Z’ (rll) The
height-height
-
foo
Z2 (rll)pl (ril, z) dz.
correlation function
(2.4)
C,, is defined by
c’o
Cz,z(rjjj)r2jj)
ZIZ2)
f
ZI Z2P2 oo
(r 111, zi, r211, Z2) dzl dZ2,
(2.5)
where zj
--
Scattering
Statistical Aspects of Wave
2
z(rjll).
Rough
at
Surfaces
63
introduce the pair-correlation function
It is also usual to
g(rlll, r2ll) which averages the square of the difference points of the surface,
height
between two
C,O
g(rill, r2ll)
-
((ZI
-
Z2)2)
(Z1 -
g(rill, r2ll)
Note that
=
Homogeneity
2.2.3
2(z 2) (rll)
-
Z2
)2 P2 (rill, zi
r2ll,
,
Z2)dz,dZ2 .(2.6)
00
-
2C,, (rill,
r2ll)
Ergodicity
and
Randomly rough surfaces have frequently the property that the character of height fluctuations z does not change with the location on the surface. More precisely, if all the probability distribution functions pi are invariant under any arbitrary translation of the spatial origin, the random process is called homogeneous. As a consequence, the ensemble average of the functional F(z, z,) will depend only on the vector difference, rj 11 r, 11 between one of the n space argument rill and the (n 1) remaining others rjll, j 2 n.
the
-
...
--
-
(F) (r 111,
...,
r,,
11)
--
(F) (0 11
...
r,,Il
-
...
(2.7)
rill).
When the random process is isotropic (i.e has the same characteristics along reduce to the distance Jrjll rilll between any direction) the dependencies will only consider hoone of the argument and the others. Hereafter we -
space
mogeneous isotropic random processes and we propose for the various functions already introduced.
The can
mean
find
altitude
reference
a
o-,
a
depend
does not
plane surface such
of the surface is also
height
(z)(rll)
constant and
z)
as
--
on
a
the rll
0. The
position
mean
define the root
we
simplified
notation and
one
square deviation
mean
square
(rms)
as
00
a2
-
(Z2)
=
f
z
2pi (z) dz.
(2.8)
00
height is often used to give an indication of the "degree of roughlarger u the rougher the surface. Note that the arguments of the probability distribution are much simpler. Similarly, the height-height correlation function can be written as,
The
rms
ness",
the
C-,, (rill,
r2ll)
-
(_z(Oll)z(rll))
-
C,,(rll)
-j
Z1 Z2P2
where rll -- jr1l 1. We also introduce, with these point and two points characteristic functions,
Xi
(8)
-
(2.9)
(Zl; Z2, r1l) dzi dZ2,
simpler notations, the
foo pj(z)e"dz,
one
(2.10)
Anne Sentenac and Jean Daillant
64
00
X2
( S SI) 7’11) )
::::::
f.
P2
(Z) z’I) r1j) e "’+""’dzdz’.
One of the most
important attributes of a homogeneous random process is its P(qjj) that gives an indication on the strength of the surface fluctuations associated with a particular wavelength. Roughly speaking, the rough surface is regarded as a superposition of gratings with different periods and heights. The power spectrum is a tool that relates the height to the period. We introduce the Fourier transform of the random variable z, power
spectrum,
I
i(qjj) where trum
qjj
--
(q, qy)
is the
=
z(rll)e iqjj.rjj dril
7r2
in-plane
wave-vector transfer. We define the spec-
as
P(qjj)
--
fli(q11) 12)
The Wiener-Kintchine theorem
[5]
Fourier transform of the correlation
P(q1j) More
(2.12)
precisely,
one
1 -_
41r 2
I dr1le
--
(,i(qjj)i(-qjj)).
(2-13)
states that the power
spectrum is the
function,
iqll.ril
(z(0jj)z(rjj))
(2.14)
shows that
?(qjj)i(q’jj))
(i(-qjj)i(q’jj))
--
--
0 , ,(q11)6(q1I
-
q’11).
(2-15)
The Fourier components of a homogeneous random variable are independent random. variables, whose mean square dispersion is given by the Fourier transform of the correlation function. If the power spectrum decreases slowly with increasing q11, the roughness associated to small periods will remain important.
Thus, whatever the length scale, the
surface will present
irregularities.
In the real space, it implies that the correlation between the heights of two points on the surface will be small, whatever their separation. As a result, the correlation function will exhibit the derivative for
function sented in
(or
exemple). spectrum)
power
Fig.
a
singular
behavior about 0
(discontinuity
of
An illustration of the influence of the correlation on
the
roughness aspect of the surface is prespecial case of a Gaussian
2.2 and detailed in Sec. 2.2.4 in the
distribution of
heights.
have been interested solely in ensemble average, which neknowledge of the complete set of rough surfaces generated by the homogeneous random process (or the probability distributions). However, sometimes only a single realisation S,, (with dimension L, Ly along Ox and Oy) of the random process is available and one defines the spatial average of z,,) for this surface by, any functional F (zi, Until
now we
cessitates the
...’
FP(011’...’r,,11)
--
lim L ,,xL,
-+oo
LLY
JLxXLy drjjF[z(rjj)
...
z(rll
+
r,,11)]
Statistical Aspects of Wave
2
Scattering
at
Rough
Surfaces
65
happens frequently that each realisation of the ensemble carries the same homogeneous random process as every other realisation. The spatial averages calculated for any realisation are then all equal and coincide with the ensemble average. The homogeneous random process is then said to be an ergodic process. In this case, the following particular relations hold, It
statistical information about the
0’
C,,(rll) One
-
2
=
(Z2)
Jim L,,,L,-+oo
(z(0jj)z(rjj))
lim Lx,L,-+oo
L,LY 1
LxLy
z2 (rjj)drjj,
(2.17)
LxxLy
j
z
T X1 Lx
(r’ll)z(rll
+
rjj)dr’1. (2.18)
show that Eqs. (2.17) and (2-18) will be satisfied if the correla(r1j) dies out sufficiently rapidly with increasing r1l (see for
can
tion function Qz z
demonstration
rough
correlated
this
property implies that
one
realisation of the
can
so
realisation.
[5]). Indeed,
be divided up into subsurfaces of smaller area that are unthat an ensemble of surfaces can be constructed from a single
surface
Spatial averaging
amounts then to ensemble
averaging.
If the
ran-
dom process is homogeneous and ergodic, all the realisations will look similar while differing in detail. This is exactly what we expect in order to describe liquid surfaces varying with time or set of surfaces of artificial origin. The fact
spatial averaging is equivalent to ensemble averaging when the surface enough correlation lengths to recover all the information about the random process is of crucial importance in statistical wave scattering theory. that
contains
The Gaussian
2.2.4
Probability Distribution
and Various Correlation Functions
height probability distribution is taken to plays a central role because it has simple structure, and, because -of the central limit theorem, it is
In most
theories,
the
The Gaussian distribution
distribution that is encountered under If the
height
whose effects altitude will
a
be Gaussian.
especially probability
an a
great variety of different conditions.
number of local independent events cumulative, (like the passage of grain abrasive) the resulting obey nearly Gaussian statistics. This result is a manifestation of
z
of a surface is due to
a
large
are
the central limit theorem which states that if
a
random variable X is the
sum
independent random variables xi, it will have a Gaussian probability distribution in the limit of large N. Hereafter, we suppose that the average 0. The Gaussian height value of the Gaussian variate z(rll) is null, (z) of N
--
distribution function is written as,
Pi
W
=
o-v’2-7r
e xp
(
-
_02
)
(2.19)
Anne Sentenac and Jean Daillant
66
Gaussian variates have the remarkable property that the random process is entirely determined by the height probability distribution and the height-
height correlation function C,,.
All
higher
[5].
in terms of second order correlation is
given P2
in this
order correlations
The
two-points
expressible
by,
case
0’
(Z, Zf, r1j)
are
distribution function
exp
IF)
2,r,
2(Z2
+
Z/2)
-
2zz’C,,, (r1j)
-
2o-
4
-
2Q,2 ,(rjj)
I
-
(2.20) Other useful results
the Gaussian variates are,
on
XI
X2
(8, S,r1j)
-
(S)
-_
(e iSZ)
=
e_S20,2/2
(2.21)
e_0,2(S2+S,2 )12,ss’C_(1jj).
(e i(SZ_31Z1))
(2.22)
The correlation function
plays a fundamental role in the surface aspect. It the length scales over which.height changes along provides the surface. It gives in particular the distance beyond which two points of the surface can be considered independent. If the surface is truly random, C,,_,(rll) decays to zero with increasing r1j. The simplest and often used form for the correlation function is also Gaussian, an
indication
on
CZZ (r1l) The correlation larities
is the
length
(or bumps)
on
-
0.2 exp(-r 2g2). 11
(2.23)
typical distance between two different irreguBeyond this distance, the heights are not
the surface.
correlated. In certain
scattering experiments,
one can
retrieve the behaviour of the
cor-
relation function for q close to zero. We have thus access to the small scale properties of the surface. We have seen that the regularity of the correlation function at
mirrors the asymptotic behaviour of the power spectrum : high-frequency components of the surface decay to zero, the
zero
the faster the
smoother the correlation function about variations about cated
solely
zero
have the
for surfaces that
zero.
The Gaussian scheme whose 2
quadratic form o- (1 (rjj / ) 2) is thus present only one typical lateral length _
indiscale
161. For surfaces with structures down to the correlation function to be
affine
rough
surface for
arbitrary small scales, one expects singular at zero. An example is the self
more
which,
g(r1j) where A0 is
a
constant,
--
2h
(2.24)
Aor 11
or
C,, (r1j)
--
o’
2
(1
r,2,
h
_
2h
)
,
(2.25)
Statistical
2
with 0 < h < 1. The
Aspects
of Wave
Scattering
at
Rough
Surfaces
67
roughness exponent or Hurst exponent h is the key paheight fluctuations at the surface: small h values
rameter which describes the
produce very rough surfaces while if h is close to I the surface is more reguh lar. This exponent is associated to fractal surfaces with dimension D -- 3 as reported by Mandelbrodt [7]. The pair-correlation function given in Eq. -
(2.24) diverges are
represented
for r1l and the
oo.
Hence, all the lengthscales along the vertical axis
roughness
of the surface cannot be defined. We will
below that in that case, there is no specular reflection. However, very often, some physical processes limit the divergence of the correlation function, see
roughness saturates at some in-plane cut-off . Such by the following correlation function,
i.e. the
surfaces
are
well
described
C, (R)
-R 2h
-
,
liquid surfaces and surfaces
For
a2exp(-
close to the
functional forms described in Chaps. 6 and 9
(2.26)
2h roughening
are
transitions other
used.
0.4
0.2
0.0
VWW
-W _V"
-5.0
0.0
-0.2
z ’31,
";s
-0.4
0-6
-0.8
-1.0
-1.2 -15.0
_10.0
5.0
10.0
X
Fig.
2.2.
Various
rough
surfaces with Gaussian
correlation functions From bottom to top,
a2
exp(-L.2-), 2
2.2.5
C,,(R)
height distribution
C,,(R)
a2 4/( 2
but various
+ R2)2, C-,, (R)
0.2 exp(-
Complicated Geometries: Multilayers and Volume Inhomogeneities More
solely the statistical description of a rough surhomogeneous media. The mathematical notions that have been introduced can be generalised to more complicated problems such as stacks of rough surfaces in multilayer components. In this case, one must also consider the correlation function between the different interfaces, (zi (011)zj (r1j))
Up
to
face
now we
have considered
separating
two
Anne Sentenac and Jean Dw.Hant
68
where zi represents the height of the ith surface. A detailed description of the statistics of a rough multilayer is given in chap. 8, Sect. 8.2. One can also
describe in
(or
a
electronic
similar fashion the random fluctuations of the refractive index
density)
p. In this
case
p is
three-dimensional space coordinates Sect. 4.5.
Description
2.3
of
a
a
random continuous variable of the
(rij, z).
Surface
It will be introduced in
Chap.
4
Scattering Experiment,
Coherence Domains
We have seen how to characterise, with statistical tools, the rough surface geometry. The next issue is to relate these statistics to the intensity scattered by the sample in a scattering experiment. In this section, we introduce the main theoretical results that describe the interaction between and surfaces. Attention is drawn
waves
which takes
on
on
electromagnetic
the notion of "coherence domains"
particular importance in the modelling of scattering from foreword, we present briefly the basic mechanisms
random media. In this
that subtend this concept.
be shown
(bear
in mind the
Huygens-Fresnel principle or see Chap. by an electromagnetic incident field acts as a collection of radiating secondary point sources. The superposition of the radiation of those sources yields the total diffracted field. If the secondary sources are coherently illuminated, the total diffracted field is the sum of the complex amplitudes of each secondary diffracted beam. In other words, one has to account for the phase difference in this superposition. As a result, an interference pattern is created. The coherence domain is the surface region in which all the radiating secondary sources interfere. It depends trivially on the nature of the illuminating beam (which can be partially coherent), but more importantly, it depends on the angular resolution of the detector. To illustrate this assertion, we consider the Young’s holes experiment [8]. Light from a monochromatic point source (or a coherent beam) falls on two pinholes located in the sample plane (see Fig. 2.3). We study the transmitted radiation pattern on a screen parallel to the sample plane at a distance D. In this region, an interference pattern is formed. The periodicity A of the fringes, which is the signature of the coherence between the two secondary AD/d. sources, depends on the separation d between the two pinholes, A Suppose now that a detector is moved on the screen to record the diffrated intensity. As long as the detector width I is close to A, the modulation of the It
can
4 Sect.
4.1.6)
that
a
rough
surface illuminated
=
interference pattern will be detected. On the contrary, if I > 10A the intensity measured by the detector is the average of the fringe intensities. We obtain a constant equal to the sum of the intensities scattered by each secondary sources.
the
In this case, one may consider that, from the detector point of view, radiate in an incoherent way. We see with this simple experiment
sources
Statistical
2
Aspects of Wave Scattering
at
Rough
Surfaces
69
length is directly linked to the finite extent of the detector angular resolution).’ to a more accurate description of a surface scattering ex-
that the coherence
(equivalent We
to
now
finite
a
turn
periment.
Scattering Geometry
2.3.1
illuminating a rough a (perfectly sample along kin monochromatic) and detecting the flux of Poynting vector in an arbitrary small solid angle in the direction k c with a point-like detector located in the far-field region. We consider
an
ideal
scattering experiment consisting
in
beam directed
coherent
with
The interaction of the beam with the material results in
a
wavevector
transfer, q
=:
k,,
-
(2.27)
kin-
qz r)lane of incidence
sample plane
Fig.
Figure
2.3.
Scattering geometry
2.3 shows the
experiment.
The
plane
for
interpreting
scattering geometry
in the
surface
scattering
general
case
of
a
of incidence contains the incident wave-vector
the normal to the surface Oz. In
surface
kij and
it is usual to work
reflectivity experiment, 0--0. Yet the case 0 :A 0 is’ of diffrac: interest for surface ,tion experiments in grazing incidence geomspecial in the of incidence it is also useful to distinguish When plane working etry. the symmetric specular geometry for which Oin Os, and the off-specular The of which set for 0ij 0,,. equations following $ (2.28) gives the geometry components of the wave-vector transfer with the notations introduced in Fig.
in the
plane
a
of incidence and thus to have
--
2.3.
obviously linked to the degree of coherence fixed by, for example, the opening. However, for x-ray or neutron experiments the resolution actually generally limited by the detector slits opening.
It is also
incidence slit is
Anne Sentenac and Jean Daillant
70
i Scattering
2.3.2
q,
=
qy
=
qz
=
cos Oin) (cos 0,, cos (cos 0,, sin (sin Os, + sin Oi )
ko ko ko
(2.28)
m
Cross-Section
experimental setup presented in the previous section, one exactly measures scattering cross-section as described in Fig. 1.1 of isolated scattering object is the rough sample in this case). The Chap. 1, (the In the ideal
the differential
vectorial electric field E is written E
as
_-
the sum,
(2.29)
Ein + Esc
plus scattered field. We are interested by the flux of the Poynta surface dS located at the position R of the detector through ing for a unit incident flux. The precise calculations of the differential scattering cross-section are detailed in Sect. 4.1-6. In this paragraph, we simply introof the incident vector S
duce the main steps of the derivation. One assumes that the detector located at R is
(far-field approximation). We define see
placed far from the sample scattering direction by the vector ks,
the
Fig. 2.3, k c
-
koil
-
(2.30)
kOR/R.
It is shown in Sect. 4.1.6 that the scattered field
can
be viewed
as
the
sum
by the electric dipoles induced in the material by the incident field, (these radiating electric dipoles are the coherent secondary
of the wavelets radiated
sources
introduction). The strength of the induced dipole sample, is given by the total field times the permittivity 2 Let us recall that for x-rays, point, [k (r’) k2]E(r). 0 in the
presented
located at r’ in the contrast at this
-
2
(k (r/)
-
k2) 0
-
2[n2(r’)
k0
-
1]
=
-47rr,p,l (r/),
(2.31)
2 where p,1 is the local electron density and r, the classical electron radius. In the far-field region, the scattered field can be written as, see Eq. (4.19), (the 2
is only interested in materials with low atomic numbers for which the frequency is much larger than all atomic frequencies, the electrons can be considered as free electrons plunged into an electric field E. In this case, the movement of the electron is governed by m,dv/dt -eE, where m, v, -e, are the mass, the velocity and the charge of the electron. We find v (ze/m,W)E for a e"t time dependence of the electric field. Thus the current density is j -(ze 2pei/m,w)E where p,1 is the local electron density. Writing the -ep,iv 2 n C0-9E/(9t Maxwell’s equations in the form curIH j + rsoaElat aDlat
If
one
x-ray
=
=
=
=
or
I
as
_
a
on
whether the system is viewed
material of refractive index
(e2 /2m,co W2)p eI=
I
-
(A2/27r)r,p,1
the "classical electron radius", A in Ref.
[91
n),
one
;:Z
complete
=
=
=
(depending
as
a
set of electrons in
obtains I
-
and
a
vacuum
by identification that
10-6,
with
r,
=
n
=
(e2 /47reo rn_ C2)
rigourous demonstration is given
Statistical
2
far-field in
approximation Chap. 4). E,;, (R)
-
Wave
Aspects of
and its
domain
validity
exp(-ikoR)
Scattering
dr’(k 2(r’)
47rR
at
discussed in
are
more
71
details
(r /) eik,,.r’
2
k 0 )E-L
-
Surfaces
Rough
(2-32)
where
Ej-(r’)
E(r’) -: i.E(r’)ii
=
(2.33)
represents the component of the electric field that is orthogonal tion of
propagation given by fi. Expression
electric field E, (R)
k,c
--
koRIR
_-
E, (k,;,)
Poynting
plane
a
wave
[8]
with wavevector
E,,c (R).
-
,
The
(2.32)
approximated by kofi and amplitude, be
can
to the direc-
shows that the scattered
(2-34)
readily obtained,
vector is then
S
I =
2poe
I E , (R) 12 ia.
(2.35)
Poynting vector for a unit incident flux (or normalized by through a unit surface normal to the propagation direction) differential scattering cross-section in the direction given by k’,
The flux of the
the incident flux
yields the
do-
1
Note that
I [k (r’)
167r2lEi,112
dQ
duldfl
involves
do-
double
a
E-L (r).E*L (r +
the vertical
over
axis,
Es, (R)
--
2
-
that k 02)
ik,,
can
dr’
(2-36)
be cast in the
(k2 (r + r)
-
obtains
of a
(2.37) By integrating formally Eq. (2-32)
u.
surface
exp(-ikoR)
S
47rR
form,
k 02)
r)e ik.,,,.r’1
conjugate
one
ko]E_L (r’)e
integration,
dr
167r2lEi,112
where u* stands for the
-
I J dr’(k (r)
I _
dQ
2 2
2
i
integral,
(r’ll, k,,c , )
ik,,:Il.r ’1
dril,
(2-38)
with S
We
see
with
that
k,,,,
hand, the nated
k,,,)
[k 2(r/)
-
k 02]eik,c, "Ei
-
(r’) dz’.
Eq. (2.39) is a ID-Fourier transform, thus the variations directly linked to the thickness of the sample. On the
are
variations of Esc with
area
(
i.e. the
region
k,;,Il
for which
are
(2.39) of Eiother
related to the width of the illumi-
[k 2(r’)
-
k2]E 0
is
non-zero).
Anne Sentenac and Jean Daillant
72
Coherence Domains
2.3.3
to now, we have considered an ideal experiment with a point-like detector. reality, the detector has a finite size and one must integrate the differential scattering cross-section over the detector solid angle, zAodet- Since the
Up In
cross-section is defined
as
function of wavevectors, it is
a
more
convenient
integration over the solid angle -60det centered about the k,, into an integration in the (k, ky) plane. The measured inten-
to transform the
direction
sity (scattering cross-section convoluted given by, I
.1
x
where
R(kil)
is then
dkjjR(kjj)
-
167r2
function)
with the resolution
jEj,, I
I drjj I dr’ll S* (rjj L I
r’ll, k.,).9j (r1j, k,)e ikjj.r’jj
+
-
is the detector acceptance in the
(k, ky) plane.
(2.40)
The expression
of R in the wavevector space is not easily obtained. In an x-ray experiment, it depends on the parameters (height, width) of the collecting slits. The reader is referred to section 4.7 for
detector
shape.
detailed
expression introductory chapter it is
In this
a
Gaussian function centered about
R (k,,,,
k,,y)
=
C exp
of R
as
a
function of the
sufficient to take for R
a
k,,,11,
(k.,
-
ks,-,)2
(ky
-
k,,,y)
2./-A k 2
2Ak2X
2
(2.41)
Y
angular aperture of the detector. If one assumes that the integrant does not vary significantly along k, inside ’ Akx 3 zAky) the resulting intensity is given by,
zAk,,zAky
The variables
I
1
1
167r2
jEj,, I
-
govern the
drIldr’119*L (rjj
+
r
/11, ksc
),ik_jj.r’jj k(r 11)
(r1j,
(2.42) where
fZ(rjj) We
now
examine
fields radiated 3
by
-
27rCzAk,zAkye
j’Ak2 XX2 _.LAh2y2 2 2 Y
(2.43)
Eq. (2.38) that gives the scattered field as the sum of the all the induced dipoles in, the sample. We see that the
assumption is not straightforward. It is seen in Eq. (2.39) that the thicker sample, the faster the variations of 9 1 with k,. In an x-ray experiment, the sample under study is generally a thin film (a couple of microns) and we are interested by the structure along z of the material (multilayers). Hence, the size of the detector is chosen so that its angular resolution permits to resolve the interference pattern caused by the stack of layers. This amounts to saying that the k,-modulation of F*L (rjj + r’ll, k-.).Ej (r 11, k_.) is not averaged in the detector. This the
-
Statistical
2
Aspects of Wave Scattering
electric field radiated in the direction ksc
n +
r
Surfaces
73
the "effective"
by
whatever the distance between the
11
Rough
dipole placed at by another dipole placed points. The intensity, mea-
to the field radiated
point rll is added coherently at
at
ideal experiment (coherent source and point-like detector), is by double a integration of infinite extent which contains the incoherent given by term 191 (r1l,kc ,)l’) and the cross-product (namely the interference term) 91 (rll, k,c,)..E*1 (rll + r’ll, k,,,,) When the detector has a fi n ite size, the dou sured
an
-
-
ble
integration
is modified
by
the introduction of the resolution function
which is the Fourier transform of the detector. In is
our
example, fZ
roughly 11[zAkx
x
contribution of the main
is
a
radiated
by
detector
(the
two
points
that
belong
to this domain will add
while the fields
important),
term value is
cross
intensity is significant. This doScoh due to the detector. The fields
term to the total
be called the coherence domain
can
characteristic function of the
Gaussian whose support in the (x, y) plane This function limits the domain over which the
Ak.].
cross
angular
fZ
coherently in the coming from two
points outside this domain will add incoherently (the cross term contribution is damped to zero). The resulting intensity can be seen as the incoherent sum of intensities that are scattered from various regions of the sample whose sizes coincide with the coherent domain given by the detector. This can be readily understood by rewriting Eq. (2.42) in the form [10], I
C)c
E i=I,N L
I
(rill
drll
Sc.,j
+ rll +
Scoh
dr’ll
r’ll, ks,,).g (rill i
+ r1l,
k,,c, )eik_jj.r’jj ’&(r’
(2.44)
regions Scoh. Hence, integrating equivalent to summing the intensities (i. e. incoherent process) from various regions of the illuminated sample. This is the main result of this paragraph. The finite angular resolution of the detector introduces coherence lengths beyond which two radiating sources can be considered incoherent (even though the incident beam is perfectly coherent). Note that the plural is not fortuitous, indeed, the angular resolution of the detector can be different in the xOy and xOz plane, thus the coherent lengths vary along Ox, Oz and Oy. In a typical x-ray experiment (see Sect. 4.7.2, the sample is illuminated coherently over 5 mm’ but the angular resolution of the detector yields coherence domains of solely a couple of square microns. More precisely, it is shown in Sect. 4.7.2 that a detection slit with height 10 mrad 100 ym width 1 cm placed at I meter of the sample with 0, c limits the coherent length along Oz to 1 ym, the coherence length along Ox to 100 ym and that along Oy to 10 nm. Finally, in this introductory section, we have restricted our analysis solely to a detector of finite extent. In general, the incident source has also a finite angular resolution. However, coherence domains induced by the incident angular resolution is usually much bigger than that given by the detector angular resolution so that we do not where the
ri
is the center of the different coherent
intensity
over a
certain solid
angle
is
--
Anne Sentenac and Jean Daillant
74
consider it here.
(The
complete description
calculation scheme would be very
of the resolution function of the
similar).
experiment
is
A
more
given
in
Sect. 4.7.2.
2.4
Statistical Formulation of the Diffraction Problem
In this
section,
point out, through various numerical simulations, the perdescription of the surface and of the scattered power for modelling a scattering experiment in which the rough sample is necessarily deterministic. The main steps of our analysis are as follow: Within the coherence domain, the field radiated by the induced dipoles (or secondary sources) of the sample interfere. We call speckle the complicated intensity pattern stemming from these interferences. The angular resolution of the detector yields an incoherent averaging of the speckle structures, (the intensities are added over a certain angular domain). This angular integration can be performed with an ensemble average by invoking 1. the ergodicity property of the rough surface, (i.e. we assume that the sample is one particular realisation of an ergodic random process) and 2. the equivalence between finite angular resolution and limited coherence tinence of
a
we
statistical
domains. is
It appears finally that the diffused intensity measured by the detector adequatly modelled by the mean square of the electric field viewed as a
function of the random variable
Throughout this section, the numerical optical wavelength is about I pyn and incident beam is directed along the Oz axis.
in the
examples given the perfectly coherent are
To What Extent is
2.4.1
z.
domain. The
a
Statistical Formulation
of the Diffraction Problem Relevant? In Sect. 2.3 it has been shown how to calculate
formally the electromagnetic by the detector in a scattering experiment. To obtain the differential scattering cross-section, one needs to know the permittivity contrast at each point of the sample, and the electric field at those points, Eq. (2.37). If the geometry of the sample is perfectly well known (i.e. deterministic like gratings), various techniques (such as the integral boundary method [11], [12]) permit to obtain without any approximation the field inside the sample. It is thus possible to simulate with accuracy the experimental results. In the case of scattering by gratings (i.e periodic surfaces) the good agreement between experimental results and calculations confirms the validity of -
power measured
the numerical simulations We
study
faces s,,
(e.g.
beam. In this
[12].
the scattered
intensity from different rough deterministic surpresented in Fig. 2.1) illuminated by a perfectly coherent experiment, we suppose that the size of the coherence domains
those
2
Statistical
Aspects ofWave Scattering
at
Rough
Surfaces
75
0-40
o.oo-9o
o
-
-70
To
.
lo
so
... cl--)
Fig. 2.4. Simulations of the differential scattering cross-section for the surfaces presented in Fig. 2.1. The illuminated area covers 40 Mm which explains the large angular width of the speckle. The incident wavelength is I Mm, the refractive index is 1.5. Normal incidence. The calculations are performed with a rigorous integral n boundary method (no approximation in solving Eq. (2.37) other than the numerical ==
discretizations) [13]
induced minated
by
the finite resolution of the detector is close to that of the illu-
area
A. In other
words,
all the
fringes
of the interference pattern
by every illuminated by the detector. We observe in Fig. 2.4 that the angular distribution of the intensity scattered by each surface presents a chaotic behavior. This phenomenon can be explained by recalling that’the scattered field consists of many coherent wavelets, each arising from a different microscopic element of the rough surface, see Eq. (2.38). The random height position of these elements yields a random dephasing of the various’ coherent wavelets which results in a granular intensity pattern. This seemingly random angular intensity behavior, known as speckle effect, is obtained when the coherence domains include many correlation lengths of the surface, when the roughness is not negligible as compared to the wavelength (so that the random dephasing amplitude is important) and most importantly when the s2ze of the coherence domains is close to that of the illmninated area so that the speckle is not averaged in the detector. To retrieve the precise angular behavior of the intensity, one needs an accurate deterministic description of the surface [14]. In Fig. 2.4 the surfaces s,, present totally different intensity patterns even though they have the same statistical properties. However, some similarities can be found in the curves plotted in Fig. 2.4. For example, the typical angular width of the spikes is the same for all surfaces. Indeed, in our numerical experiment it is linked to the width L of the illuminated area (which is here equivalent to the coherence domain). The smallest angular period of the fringes formed by the (farthest-off) coherent point-source pair on
stemming
from the coherent
point of the surface
are
sum
resolved
of the fields radiated
Anne Sentenac and Jean Daillant
76
angular width A/L of the speckle spikes. Fig. 2.5, the larger the coherently illuminated angular speckle structures. In optics and radar imaging,
the surface determines the minimal
This is area
clearly
illustrated in
the thinner the
sufficiently coherent incident beams (lasers) combined with detectors with fine angular resolution permits to study this phenomenon [14]. In x-ray experiments, the speckle effect can also be visualised in certain configurations. I mrad), the apparent resolution of the detector At grazing angles (e. g. 0,, Sect. k00S0 Jq,, 4.7.2) may be better than 10-7 ko in-’. The size of (see the illuminated area being 5 mm, the speckle structures are resolved in the --
--
detector.
0.6
0.4
li,93
0.2
0.0 -so
-30
_10 ()
Fig.
2.5. Illustration of the
dependence
tures on the size of the illuminated
bution for case over
one
rough
10
_
area.
30
50
(Clegree.)
of the
angular
width of the
Simulation of the
surface illuminated in the first
30pm. The incident wavelength is I
specIde strucintensity angular distri-
case over
60 tim and in second
pm, the refractive index is
n
=
1.5,
normal incidence
now suppose that the illuminated area is increased enough so that typical angular width of the speckle structures will be much smaller than the angular resolution of the detector. The detector integrates the intensity over a certain solid angle and, as a result, the fine structures disappear. One notices then that the smooth intensity patterns obtained for all the different surfaces s,, are quite similar. This is not surprising. Indeed, we have seen in the previous paragraph that the finite angular resolution of the detector is equivalent to the introduction of a coherence domain Scoh (that is smaller than the illuminated area A). The measured intensity can be considered the incoherent sum of intensities stemming from the different subsurfaces of size Scoh that constitute the sample. We now suppose that the illuminated area is big enough to cover many "coherent" subsurfaces, A > 30Scoh. Moreover, we suppose that the coherence domain is large enough so that each subsurface presents the same statistical properties Lcoh > 30
We
the
Statistical
2
Aspects of Wave Scat ering
at
Rough
Surfaces
77
length and Lcoh the coherence length. If the set of by an ergodic stationary process, the ensemble of subsurfaces obtained from one particular realisation sj will define the same random process with the same ensemble averaging as that created from any other realisation sl,. Consequently, the scattered intensity from one "big" where
is the correlation
surfaces
f s,}
be described
can
surface sj can be seen as the ensemble average of the "subsurface" Sc,,h scattered intensity which should"be the same for all sk. This assertion is
supported by a comparison between two different numerical the’same scattering experiment [13] [15].
treatments of
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0 -90
-30
-60
Fig. 2.6.
Simulation of the differential
3mm
(roughly
one
scattering cross-section
realisation of
M. Saillard
In
Fig. 2.6 rough
ministic with
a
of
a
rough
deter-
a
random process are: Gaussian height distribution correlation function with Ipm. The incident
prof.
90
60
(Clegree)
random process. The illuminated area several thousand of optical wavelengths). The statistics of the
ministic surface which is covers
30
0 0
with, o0.2pm and Gaussian wavelength is lpm. Courtesy of =
[13]
we
have
surface
plotted
Sj
intensity obtained from a deterby a perfectly coherent Gaussian beam, The rough surface is one realisation of a
the diffuse
illuminated
detector of infinite resolution.
random process with Gaussian height distribution function and Gaussian correlation function with correlation length . The incident beam is chosen wide
enough
so
that the illuminated part of
Sj
is
representative
of the
ergodic
random process. In other words, Sj can be divided into many subsurfaces (with similar statistical properties) whose set describes accurately the ran-
dom process. The total length of the illuminated spot is 5000 . It is seen Fig. 2.6 that the scattered intensity exhibits a very thin speckle pattern.
in
Fig. 2.7 we have averaged degree, corresponding to the angular resolution of a detector. We compare in Fig. 2.7 the angular averaged pattern with the ensemble average of the scattered intensity from subsurfaces In
general
the diffuse
these fine structures
intensity
over an
are
not visible. In
angular
width of 5
Anne Sentenac and Jean Daillant
78
0.9
0.8
0.6
0.5
0.3
0.2
0.0 -84
;6
-56 0
Fig.
4
&Iegree )
Solid line:
Angular average over 5 degrees of the differential scattering ’big surface’ presented in Fig. 2.6, dotted line: ensemble average differential scattering cross-section of rough surfaces with the same statistics "big surface". Size of each realisation is 30pm, no angular averaging. Courtesy
2.7.
cross-section of the of the as
the
of Prof. Saillard
that
are
[13]
generated
with the
illuminated domain is induced
by
now
same
random process
as Sj but whose coherent (i.e. to the coherence domain detector). We obtain a perfect agree-
restricted to
the finite resolution of the
30
ment between the two
scattering patterns. In this example, we do not need precise value of the characteristic function z(rll) but solely the statistical properties of the random process that describes conveniently these particular surfaces. The integration of the intensity over the solid angle AQ will then be replaced by the calculation of the ensemble average of the intensity. This ensemble averaging appears also naturally in the case of surfaces varying with time (such as liquid surfaces like ocean) by recording the intensity during a sufficiently long amount of time. any
the
longer
Each subsurface (either spread spatially temporally) generates an electric field E. The
via the coherence domains latter
can
be viewed
as a
or
func-
tion of the random process z. The intensity measured by the detector is then mean (in the ensemble averaging sense) square of the field,
related to the
(JE 12
.
The purpose of most
ious moments of E. More mean
and
a
wave scattering theories is to evaluate the varprecisely, the random field can be divided into a
fluctuating part,
E
We
usually study separately the
--
(E)
+ JE.
different contributions to the
(2.45)
intensity.
Statistical
2
Notions
2.4.2
on
Aspects of Wave Scattering
Rough
Surfaces
79
(Specular) (Diffuse) Intensity
Coherent
and Incoherent In the
at
the scattered electric field E,, behaves like a plane wave with ks, and amplitude E(k,,) see Eq. (2.32). It can be written as the mean part and a fluctuating part,
far-field,
wavevector
of
sum
a
Esc
=
(E ,)
(2.46)
Es_
+
previous discussions have shown that the measured scattered intensity a rough sample (whose deterministic surface profile is assumed to be one realisation, of a given ergodic random process) can be evaluated with the ensemble average of the intensity JE,;,(k c) 12), The
from
(JE,, 12) The first term
intensity
on
the
right
--
I (E,,) 12
hand side of
(IJE" 12).
+
equation (2.47)
while the second term is known
as
(2.47) is called the coherent
the incoherent
intensity.
It is
sometimes useful to tell the coherent and incoherent processes in the scattered intensity. In the following, we show that the coherent part is a Dirac function
solely to the specular direction [4] if. the randomly rough statistically homogeneous in the (Oxy) plane. In most approximate theories, the random rough surface is of infinite extent and illuminated by a plane wave. Suppose we know the scattered far-field E,,c from a rough surface of defining equation z z(rll). We now address the is shifted horizontally by whole the surface when modified of how is issue E, not such shift will d. is clear that It a a vect’or modify the physical problem. the incident However wave amplitude acquires an additional phase factor and similarly each scattered plane wave Esc acquires, when reexp(ikin.d) the primary coordinates, the phase factor exp(-ik,d). Thus we turning to obtain, that contributes
surface is
=
z(r I, -d)
-i(k_-kj ,).d E Sz (-.11) C
ESC
(2,48)
suppose that the irregularities of the rough surface stem from spatially homogeneous process. In this case, the ensemble average invariant under any translation in the (xOy) plane.
We
now
random
(Ez(rll -d) SC
This
equality
is
only possible
(E"(ni)) Sc
a
is
(2.49)
if
(E,,)
--
AJ(k,cll
-
ki,11).
(2.50)
Hence, when the illuminated domain (or coherence domain) is infinite, the coherent intensity is a Dirac distribution in the Fresnel reflection (or transmission) direction. For this reason it is also called specular intensity. Note
Anne Sentenac and Jean Daillant
80
that unlike the coherent term, the incoherent intensity is a function in the k"cll plane and its co’ntribution in specular direction tends to zero as the detector
acceptance is decreased. In’real life, the incident beam is space-limited, the finite, thus the specular component becomes a function whose angular width is roughly given by A/Lcoh. coherence domain is
solely interested in the specularly intensity. configuration allows the determination of the zdependent electron density profile and is often used for studying stratified interfaces ( amphiphilic, or polymer adsorbed film). The modelisation of the coherent intensity requires the evaluation of the single integral Eq. (2-32) that gives the field amplitude while the incoherent intensity requires the evaluation of a double integral Eq. (2.37). It is thus much simpler to calculate only the coherent intensity and many elaborate theories have been devoted to this issue [4]. Chapter 3 of this book gives a thorough description of the main techniques developed for modelling the specular intensity from rough multilayers. However, it is important to bear in mind that the energy measured by the detector about the specular direction comes from both the coherent and incoherent processes inasmuch as the solid angle of collection is non zero. The incoherent part is not always negligible as compared to the coherent part especially when one moves away from the grazing angles. An estimation of both contributions is then needed to interpret the data. In many x-ray
reflected
2.5
experiments,
one
is
This
Statistical Formulation of the Scattered Under the Born
Intensity
Approximation
section, we illustrate the notions introduced previously with a widely used model that permits to evaluate the scattering crosssimple section of random rough surfaces within a probabilistic framework. We discuss the relationship between the scattered intensity and the statistics of the surfaces. The main principles of the Born development have been introduced in Chap. 1, Appendix LA, and a complementary approach of the Born approximation is given in Chap. 4 with some insights on the electromagnetic properties ofthe scattered field. In this last
and
2.5.1
The Differential
Scattering
Cross-Section
Eq. (2.32) that gives the scattered far field as the sum of the by the induced dipoles in the sample. The main difficulty of this integral is to’evaluate the exact field E inside the scattering object. In the x-ray domain, the permittivity contrast is very small 10-’) and one can assume that the incident field is not drastically perturbed by surrounding radiating dipoles. Hence, a popular assumption (known as the Born approximation ) is to approximate E by Ein With this approximation the integrant We start from
fields radiated
-
Statistical
2
Aspects
of Wave
Scattering
Rough
at
Surfaces
81
readily calculated. For an incident plane wave Eirle- iki,,.r , the differential scattering cross-section can be expressed as, is
do-
_1
M
167r2
I Eil _L 12 lEi,,12
I 1 dr’[k 2(r) dr
projection of the
where Ei,,j_ is the
-
k 02] [k 2 (r’)
-
k2]eiq.(r-r’)’ 0
incident electric field
on
the
to the direction of observation of the differential cross-section.
unit vectors in direction Ei,, and E,;,
)’ in
--
Ein/Ei,,
and
2
(’ c
(2.51)
plane normal Denoting the
E,,/E,,
-
re-
Ein ( iin . sc)2. In x-ray experiments, the incident spectively, we have I Ei,, I field impiges on the surface at grazing angle and one studies the scattered intensity in the vicinity of the specular component. In this configuration, the orthogonal component of the incident field with respect to the scattered direction is close to the total incident amplitude. Yet, we retain the projection in the differential scattering cross-section for completeness term and coherence with the results of Chap. 1. Bearing in mind the value of the permittivity contrast as a function of the electronic density, Eq. (2.31), equation (2.51) simplifies to, dor
2(; in. sc)2
dr’pel (r)pe (r’) e iq.(r-r’)
dr
e
(2.52)
with p.1 the electron density and r, the classical electron radius. ’ In the case of a rough interface separating two semi-infinite homogeneous media one gets,
do-
2
dQ
2
r. p. 1
-
(ein.e, c)
Integrating Eq. (2.53) term to
ensure-
2
100
over
2
Pei 7e,
dQ
q2
can
z(rli)
z
r
/11 dz’
dz -00
J dr11 J dr’l
(z, z’) (with the inclusion -oo) yields,
of
a
Ie
iq.(r-r’
small
(2.53)
absorption
the convergence at
do-
One
2
( in- isc )2
Z
J dr11 I dr’l
1e
iqll. (r11
-
r’11) eiq., [z (r11)
-
z
(r’11)].
(2.54)
general presentation of elastic scattering under the Born, approxscattering by an isolated object as presented in Qhap. I Sect. appendix I.A. The differential scattering cross-section can be cast in
make
a
imation from the 1.2.4 and
the form do"
dQ
be iq.rj
2
2
drpbe
iq.r
where p is the density of scattering troduced in Eq. (1.34). The complex
objects and b their scattered length as inexponential is the result of the phase shift between waves scattered in the ksc direction by scatterers separated by a vector r as shown in figure 2.8. For neutrons, b is the scattering length which takes into account the strong interaction between the neutrons and the nuclei ( we do not consider here magnetic materials); for x-rays, b r, (e2/47reo M’_ C2) 2.810-15m =
which is the classical radius of the electron.
=
=
Anne Sentenac and Jean Daillant
82
q
--------------
-
kin
k q
sc
r
Fig.
Phase shift between the
2.8.
rated
by
a
vector
r.
The
phase
waves
shift is
scattered
(k,,
-
by
ki,,).r
=
two
points
scatterers sepa-
q.r
equation concerns a-priori the scattering from any (deterministic or not) object. In this chapter, we are mostly interested by the scattering from surfaces whose surface profile z is unknown or of no interest. We have seen in the preceding sections that if z is described by a random homogeneous ergodic process, the intensity measured by the detector can be approximated by the ensemble average of the scattering cross-section. It amounts to replacing in Eq. (2.54) the integration over the surface by an ensemble average, f f (q) dr11 L., Ly (f , where L,,, Ly are the dimensions of the surface along Ox and Oy. One obtains, This
-_
P 2r2 el 6 LxL Y
do-
(6in sc)
2
-
q2
dQ
dr1le iqll.rll ( e
iq,
[z(r11)-z(011)]
(2.55)
z
Note that the
expression (2.55)
of the differential
scattering cross-section Hence, this integral
accounts for both the coherent and incoherent processes.
does not converge in the function sense, it contains a Dirac distribution if the surface is infinite. This property will be illustrated with various examples in the
following.
If the
probability density
of
z
is
Gaussian,
we
can
write the
differential cross-section as,
do-
2r e2LxL Pei
-
2
dQ
We
see
tering)
that,
qz
Y(-eir,-i ,,) 2
fdrp
iq11.r11 )e-
5q.2,([z(rll )_Z(011)]2) 2,
(2.56)
approximation (where we neglect multiple scatintensity is related to the Fourier transform of the ex-
under the Born
the scattered
pair-correlation function, g(r11) [z(rll) Z(011)]2). In the by studying the differential scattering crosssection for various pair-correlation functions. We start by the expression of the scattering differential cross-section in the case of a flat surface.
ponential of following we
the
illustrate this result
=
_
Statistical
2
ideally flat surfaces g(r1j) tering cross-section yields :
is
For
2
dQ
qZ
2
dS?
q,
is thus
intensity
expected,
for
a
1 drjj iqjj.rjj.
constant
a
2 2 L ,L Y 47r 2 r.p.
do-
direction. As
2
(; i.
is the Fourier transform of
The scattered
Surfaces
Rough
at
83
at the surface and the scat-
everywhere
2
2
-
integral
zero
r,p,,LL Y
do-
The
Scattering
Surfaces
Ideally Flat
2.5.2
of Wave
Aspects
(2.57)
that,
so
5
2
(2.58)
(; in .;&Sc) J(qll).
Dirac distribution in the Fresnel reflection
a
perfectly
surface, the reflectivity comes solely scattering is null (JE 2)
flat
a coherent process (Sect. 2.4.2), 0. Note that the reflectivity decreases
the incoherent
from
to the
more
a
complicated problem an homogeneous ergodic
described statistically by Self-Affine
2.5.3
-
power law with q,. We now turn of scattering from rough surfaces that are as
Rough
random process.
Surfaces
Surfaces Without Cut-Off We first consider self-affine
rough
surfaces with
2h pair- correlation function g given by Eq. (2.24), g(r1j) -- Aor 11 With this paircorrelation function, the roughness cannot be determined since there is no *
scattering
saturation. The
2
cross-section is in this case,
2
r. p., L,, L Y
do-
2
dQ
and
in
expressed
be
can
qZ
( &in . SC)2
polar coordinates
2
ro p2, L, Ly
do-
( &in isc)
-
2
dQ
2
-
qZ
g12 -1 A R2h
I drlle-
1
eiqjj.rjj
(2.59)
Jo (qjj r1j),
(2.60)
as, 2
"f AR 2h
drii e-
2
with q1I being the modulus of the in-plane scattering wave-vector, and J0 the zeroth order Bessel -function. The above integral has analytical solutions for 1 and has to be calculated
h
0.5 and h
h
1, the integration yields,
--
2
2
r. p., L, Ly
do-
2
dQ
and for h
-
dQ
is
purely Let
us
2
in other
4
SC)2e-qII/q.
cases.
For
(2.61)
0.5, dd-
The above
qz
(-&in .
numerically
r
(; in . &sc)2.
2
2
7rA
ePel LL Y 2
qz
(qjj
2
+
(A) 2
2
q4 Z
)3/2
(2.62)
expressions clearly show that for surfaces of this kind the scattering (no Dirac distribution, no specular component).
diffuse
recall that
6(qll)
=
=4,1 f e-iqll -’H dr1j. v
Anne Sentenac and Jean Daillant
84
Surfaces with Cut-Off
Rough
surfaces
when the correlation function C.,,,
(r1j)
are
said to present
tends to
a
when r1l
zero
cut-off
length increases, (for
2h
2 o example see Eq. (2.26), when C,,, (rj() exp J,-h ), the cut-off is ). In this general case an analytical calculation is not possible and the scattering cross-section becomes,
do-
r
2 6
2
Pel L , L Y
q 20,2
2
dQ
qz
gr’j’ SC)2
I driie
integrant in Eq. (2.63) does not tend to integration over an infinite surface does not exist The
do-IdO
q.2C_
(r1j) iqjj.rjj
(2-63)
0 when r1l is increased. The in the function sense. Indeed,
accounts for both the coherent and incoherent contributions to the
possible extract the specular (coherent) and (incoherent) compone4ts by writing the integrant in the form,
scattered power. It is
e
(or
The distributive part
ular
reflectivity
then cast in the
q ,C_(rjj) Dirac
while the
I +
=
(C q.2,,C,_(rjj) 1)
the diffuse
(2.64)
-
function)
.
characterises the coherent
regular part gives
the diffuse power.
or
spec-
Eq. (2.63)
is
form,
( )
do-
do-
-
dQ
dQ
dodQ
coh
)
(2-65) incoh
with
(M) do-
r
2
2
Pei LL Y
6
-
q.2Cr2
( &i. - sc)
drile iqll.ril
2
-
qz
c.h
2 2 4 7r 2 repelL
,Ly
q2
2
qz
U2
J(qll ) ( &,nj sc) 2,
(2-66)
and d t7 M
r
2 e
p
e2’l L,, LY
-
,
q 2U2
2
qz
incoh
-
&Sc)
2
drjj
e
q.2, C,,,, (r1j) -1
e
iqll.r1j.
(2-67) specular part is similar to that of a flat surface except that it is reduced 2’ The diffuse scattering part may by roughness Debye-Waller factor e-q be determined numerically if one knows the functional form of the correlation function. When qZ2C (r1j) is small, the exponential can be developed as I + qZ2C (r1j). In this case, the differential scattering cross-section appears to be proportional to the power spectrum of the surface P(qjj), The
2
the
.
Z
,
_
Zz
dodQ
)
-
incoh
r
2P 2jLj Y e- q 2 0,2 47r2p (qj I) ( iin jl sc) 2.
6
e
(2.68)
2
Statistical
Aspects of
Wave
Scattering
at
Rough
Surfaces
85
Eqs. (2.66, 2.68) that the Born assumption permits to evaluate scattering cross-section of rough surfaces in a relatively simple way. This technique can be applied without additional difficulties to more complicated structures such as multilayers or inhomogeneous films. Unfortunately, in many configurations, the Born assumption proves to be too restrictive and one can miss major features of the scattering process. More accurate models such as the Distorted-wave Born approximation have been developed and are presented in Chap. 4 of this book. Yet, the expressions of the coherent and incoherent scattering cross-se(Aions given here given by the first Born approximation provide useful insights on how the measured intensity relates to the shape (statistics) of the sample. The coherent reflectivity, Eq. (2.66), does not give direct information on the surface lateral fluctuations, except for the overall roughness o-, but it provides the electronic density of the plane substrate. Hence, reflectivity experiments are used in general to probe,, along the vertical axis, the electronic density of samples that is roughly homogeneous in the (xOy) plane but varies in a deterministic way along Oz (e.g. typically multilayers). Chapter 3 of this We
see
with
both the coherent and incoherent
book is devoted to this issue. On the other hand the incoherent
scattering height-height correlation function of the surface. Bearing in mind the physical meaning of the power spectrum, Sect. 2.2.3, we see that measuring the diffuse intensity at increasing q1I permits to, probe the surface state at decreasing lateral scales. Hence, scattering experiments can be a powerful tool to characterise the rough sample in the lateral (Oxy) plane. This property will be developed and detailed in Chap. 4. Eq. (2.68) is directly linked
to the
References. Spizzichino. The scattering of electromagnetic waves from rough surfaces. Pergamon Press, Oxford,UK, (1963). F. G. Bass and 1. M. Fuks. Wave -scattering from statistically rough -surfaces. Pergamon, New York, (1979). J. A. Ogilvy. Theory of wave scattering from random rough surfaces.’Adam Hilger, Bristol,UK, (1991). G. Voronovich. Wave scattering from rough -surfaces. Springer-Verlag, Berlin,
1. P. Beckmann and A.
2.
3.
4.
(1994). Optical coherence and quantum optics. Cambridge University Press, Cambridge USA, (1995)C. A. Gu6rin, M. Holschneider, and M. Saillard, Waves in Random Media, 7,
5. L. Mandel and E. Wolf.
6.
331-349,(1997). 7.
B.B.
Mandelbrodt,
"
The fractal geometry of
nature", Freeman, New-York
(1982). Principle of Optics. Pergamon Press, New York, (1980). Oxtoby, F. Novack, S.A. Rice, J. Chem. Phys. 76, 5278 (1982). 2740 (1998) S.K. Sinha, M. Tolan and A. Gibaud, Phys. Rev. B 57 M. Nieto-Vesperinas and J. C. Dainty. Scattering in Volume and Surfaces. Elsevier Science Publishers, B. V. North-Holland, (1990).
8.
M. Born and E. Wolf.
9.
D.W.
10. 11.
,
Anne Sentenac and Jean Daillant
86
12.
13. 14.
15.
R.
Petit, ed. Electromagnetic Theory of Gratings. Topics in Current Physics. Springer Verlag, Berlin, (1980). M.’Saillard and D. Maystre, J. Opt. Soc. Am. A, 7(6), 982-990, (1990). J. C. Dainty, ed. Laser speck* and related phenomena. Topics in Applied physics. Springer-Verlag, New York, (197151). M. Saillard and D.
Maystre, Journal of Optics 19, 173-176, (1988).
Specular Reflectivity and Rough Surfaces 3
from Smooth
Alain Gibaud Laboratoire de
Physique
Condens6, UPRESA 6087 9, France
de I’Etat
,
Universit6 du
Maine Facult6 des sciences, 72085 Le Mans Cedex
It is well known that
is reflected and transmitted with
direction of
an
ent
light propagation at
optical properties. The
to observe in the visible
used
(see
a
change
in the
interface between two media which have differ-
effects known
the introduction for
reflection and refraction
as
are
easy
difficult when x-ray radiation is historical presentation). The major reason
spectrum but a
more
for this is the fact that the refractive index of matter for x-ray radiation does not differ very much from unity, so that the direction of the refracted beam does not deviate much from the incident
The reflection of x-rays
one.
is however of great interest in surface science, since it allows the structure of the uppermost layers of a material to be probed. In this -chapter, we present
general optical rough surfaces and
the
3.1
The
from 3.1.1
It
was
is
(see
formalism used to calculate the
reflectivity
interfaces which is also valid for
x-rays.’
of smooth
or
Reflected’Intensity an Ideally Flat Surface
Ba.sic Concepts shown in
Chap.
Sect. 1.4.2 for
1 that the refractive index of matter for x-ray radiation more
details): n
The classical model of
pression
an
=
I
-
J
-
i,3.
bound electron
elastically
(3.1) yields the following
ex-
of J:
6
A2 ‘=
21r
reN7
(3.2)
where re is the classical electron radius (r, 2.810-5A), A is the wavelength and p, is the electron density of the material. This shows that the real part =
The basic concepts used to determine the reflection and transmission coefficients of an electromagnetic wave at an interface were first developed by A. Fresnel [1] in his mechano-elastic
theory
of
light.
J. Daillant and A. Gibaud: LNPm 58, pp. 87 - 120, 1999 © Springer-Verlag Berlin Heidelberg 1999
Alain Gibaud
88
of the refractive index
mainly depends on the electron density of the material 10-6 and 13 is ten on wavelength. Typical values for J are 10-5 times smaller. A similar equation holds for neutrons where r,p, has to be replaced by pb (see Chap. 5, Eq. (5.24)). A specific property of x-rays and neutrons is that since the refractive index is slightly less than 1, a beam impinging on a flat surface can be totally reflected. The condition to observe total external reflection is that the angle of incidence 0 (defined here as the angle between the incident ray and the surface) must be less than a critical angle 0,. This angle can be obtained 1, yielding in absence of by applying Snell- Descartes’ law with COS Otr absorption: and
the
_
-
cosO,
-_
n
I
-_
-
(3.3)
J.
10’, the critical angle for total external reflection clearly extremely small. At small angles, COO, can be approximated as
Since J is of the order of is I
-
OC/2
and
(3.3)
becomes
02
-
c
The total external reflection of
an
(or neutron)
x-ray
grazing angles of incidence reflectivity decreases very rapidly
(3.4)
2J.
only larger angles,
beam is therefore
observed at
below about 0 < 0.5’. At
the
as
mentioned above.
chapter, we will calculate the reflectivity as a function of the 47r sin O/A incident angle 0 or alternatively as a function of the modulus q of the wave-vector transfer q (see Eq. (2.28) and Fig. 2.3 with 0s, 0). This means that the following ratios, In this
=
=
R(O)
R(q)
1_(0)
(3.5)
-
10
=
I(q)
(3-6)
10
determined, where 1(0) or I(q) is the reflected intensity (Flux of Poynting’s vector through the detector area) for an angle of incidence 0 (or wavevector transfer q), wrid 10 is the intensity of the incident beam. The theory of x-ray reflectivity is valid under the assumption that it is possible to consider the electron density as continuous (see Chap. 1). Under this approximation, the reflection is treated like in optics, and the reflected amplitude is obtained by writing down the boundary conditions at the interface, i.e., the continuity of the electric and magnetic fields at the interface, leading to the will be
classical Fresnel relations.
3.1.2
Fresnel
Reflectivity
The reflection and transmission coefficients
conditions of continuity of the electric and
can
be derived
magnetic
by writing
the
fields at the interface.
Specular Reflectivity fi,oni Smooth
3
Rough
and
Surfaces
89
intensity, which is the square of the modulus of the reflection quantity measured in an experiment. Let us consider an coefficient, electromagnetic plane wave propagating in the xOz plane of incidence, with its electric field polarised normal to this plane along the Oy direction. The 0 as interface between air and the reflecting medium which is located at z better order to In be assumed to will be in 3.1 emphasize abrupt. shown figure that the same formalism applies for x-rays and visible optics we use in this section the angles defined from the surface normal as in optics, together with the grazing angles usually used in x-ray or neutron reflectivity. The reflected
is the
--
AZ
X
0 n=]
n=1-840
Fig. 3. 1. travelling
k
Reflection and refraction of in the xOz
plane
an
incident
wave
polarised along
y and
of incidence
expression for the electric field in a homogeneous medium is derived equations which when combined, lead to the propagation field known as Helmholtz’s equation (see Chap. 1, electric the of equation for details) Eqs. (1.12), (1.15) The
from Maxwell’s
kj2E
, AE + where
kj
-
0,
(3.7)
is the wave-vector in medium 1. The electric field which is solution of equation is given for the incident (in), reflected (r) and transmitted
Helmoltz’s
(tr) plane
waves
by,
Ej
=
Aje i(wt-kj.r),6Y
(3.8)
jk,j 27r/A jkt,J /n, and,&y is a unit vector 1ki,,j withj=in, r or tr, ko along the y axis (see figure 3.1). Note that the convention of signs used in crystallography is adopted here (see Part 1, Chap. 1, by F. de Bergevin for details). It is straightforward to show, that the components of the(in), (tr), =
and
(r)
wavevectors are,
=
=
--
Alain Gibaud
90
kin kr ktr The
=
--
=
tangential component
interface fields.
(z
=
0).
In
ko (sin ii 6, ko sin ij,&, + kon (sin i2 6x
that the medium is
Assuming completely absorbed,
beam to be
Aine i(wt-ko
sin
COS
COS
-
(3-9)
i2 6z
of the electric field must be continuous at the
the field is the
air,
ii,&,) il,&,,)
COS
-
the
sum
sufficiently thick following relation
ijx) + A, i(wt-kosinijx)
Equation (3.10) must be valid for hold,
of the incident and reflected
for the transmitted must be
fulfilled,
sin i,2x) At, i(wt-kon
--
any value of x,
so
that the
(3-10)
following
con-
dition must
sin il
This condition is result of
this,
--
n
Sin i2
simply the well-known Snell- Descartes’ second law. As a perpendicular component of the electric
the conservation of the
field leads to,
Ain + A, It will be assumed that the media
--
At,.
(3-12)
non-magnetic
are
so
that the
tangential
component of the magnetic field must also be continuous. According to the
Maxwell-Faraday equation,
17
the
x
tangential component Bt i.e.,
the unit vector 6,,
E-
OB -
_W
is the dot
(3.13)
-iwB
-
product
of the
magnetic
,
X
Bt
(3-14)
2W
Since the electric field is normal to the incident the y axis and the curl of the field
17
The
field with
tangential component
x
E
-
of the
plane,
it is
polarised along
gives,
OEy6z ax
_
magnetic
OEy6 ’. az field is then
(3-15) given by
Specular Reflectivity
3
Bt
from Smooth and
I
My
iW
C9Z
1 +
r
--
t
I
r
--
nt
-
by the
use
case
of
an
COS
i2
COS
Z1
reflected
COS
(3-17)
i2
and the transmitted
one
t
(3.18)
amplitude coefficient
COS
il
-
n COS
i2
COS
ii +
n COS
i2
in the
case
(3-19)
of the Snell- Descartes’ relation leads to,
Sin(i2 sin(ij
r
In the
nAt,
--
Combining these two equations, the of a (s) polarisation is found to be,
which
--
A,/Ain amplitude r following relations are obtained,
the reflected the
91
(3.16)
-
(Ain -A,) COS ii At,/Ain,
Surfaces
easy to show that the conservation of this
and from equation (3.10) it is quantity yields,
Writing
Rough
electric field
parallel
-
+
il) i2)
to the
(3.20) plane
of
incidence,
a
similar
calculation leads to,
r
tan(i2 tan(i2
(P)
-
+
il) il)
(3.21)
equations are known as the Fresnel equations [1]. It is easy to show r(’) -- r. Only (8) that at small grazing angles of incidence for x-rays r(P) the to plane of incidence) polarisation (electric field polarised perpendicular also be given for (p) will results will be considered in detail below but some
Those
polarisation)
-
grazing angle of incidence 0 that the incident beam makes with the reflecting surface is usually the experimental variable in a reflectivity measurement. It is therefore important to express the coefficient of reflection as from a function of this angle 0 and also of the refractive index n. Starting The
COS
il
COS
il +
-
n COS
i2
n COS
Z2
(3.22)
Alain Gibaud
92
using the fact that the 0 and il, and the Ot, angles as shown in Fig. 3.1, Eq. (3.22) becomes: and
sin 0
-
sin 0 +
Applying
the Snell- Descartes’
the
following
In the
case
n
sin
complementary
(3.23)
Otr’
sin 0
Otr
n cos
=
(3.24)
reflection,
v n_2
-
-
cos2 0
sin 0 +
V1n_2
(3.25)
angles (for
*
-
cos2 0
which cosO
=
for which the refractive index
2
1
--
-
2S
1
_
(in
02 /2) and for the absence of
02C.
1
(3.26)
general equation Eq. (3.25) becomes,
r
The is
n
are
_
of small incident
electromagnetic x-ray waves absorption) is given by,
The
0
coefficient of
(0)
sin Otr
i2
law, cos
produces
n
and
reflectivity given by,
if the
-
-
0+
VF02 VO-12
02 -
(3.27)
02C
which is the square of the modulus of the reflection
R
Finally,
0
(0)
(0)
absorption
the refractive index takes
=
r
r*
O
-
0 +
V/0 2 Oc2 V O2 C
2
coefficient,
(3.28)
of the x-ray beam by the material is accounted for, a complex value and the Fresnel reflectivity is then
written,
R
The q:
reflectivity
can
(0)
--
rr*
0
-
--
equally well
V02
0 +
be
given
-
02C
-
2i/3
_022i)3
(3.29)
C
in terms of the wave-vector transfer
from. Synooth and
Specular Reflectivity
3
Rough Surfaces
93
2
R (q)
q,
q2z-
-
2
32iir2o
2
32i7r2o
qc
>’2
(3.30)
=
q, +
Vq,
2
,
q,
\
2
compared
When the wave-vector transfer is very large following asymptotic behaviour is observed:
to qc
i.e. q >
3q, the
4
qc
R
(3.31)
16q4
101
5: 1071.
q,;
P 0 LU
107’
Fresnel
--I
Reflectivky Reflecfivky Refleowky
U_
LIJ 10, cl
4
6q /16%4 qc /I 4
z
LLJ
107’.
01075
3qc
< 0.00
0.
0.’10
0.65
%
Fig.
It
can
profile -
for
3.2. Calculated
a
be
seen
reflectivity
from the
of
a
0. O
5
0.25
(A ’) flat silicon wafer and
Fig. 3.2, that the reflectivity regimes:
asymptotic law
curve or
reflectivity
(see
Sect. 7.1.1
consists of three different
plateau of
more
total external reflection R
=
I when q < q,
details)
-
a
very
steep decrease when
q
-
a
11q’
power law when q >
3q,.
--
q,
noting that if the value of q, is measured experimentally, this immediately yields the value of the electron density in the material (see Part 1, Chap. I by F. de Bergevin) since,
It is worth
q,
=
3.75 10-2 \’F P e"
(3.32)
,
where p, is the electron density in the units Finally remembering that the reflectivity is observed under
tions, reference to the system of axes defined in Fresnel reflectivity R(q) can be written as:
Fig. 3.1,
specular
condi-
shows that the
Alain Gibaud
94
2
q,
R(q)
-
qz +
q2 2
qz
-
q2
_
32i7r2p
C
-
2
qC
-
(3-33)
q,,Jqy,
222N-5-21 .X2
equation (3.33), and the reflectivity of a flat surface is only specular direction. Equation (3.33) completely describes the reflectivity of a homogeneous material, showing in particular that the reflectivity differs from zero only for wave-vector transfers normal to the surface of the sample. 2 Figure 3.2 illustrates the calculated reflectivity curve for a silicon wafer in the power law regime and also in the case of a more complete dynamical calculation. The deviation from unity due to the absorption of the x-rays in the material can be seen to play a major role in determining the form of the curve in the region close to the critical edge at q q,. Equation (3.33) shows quite clearly that the calculation of a reflectivity curve requires only the electron density and the absorption of the material (for the wavelength used). Table 3.1 gives some useful data for calculating the reflectivity of various elements and compounds. A much wider data base of quantities relevant to reflectivity measurements can be found at the following web site, "http: / /www-cxro.lbl.gov/optical-const ants/". As a conclusion of this section we wish to stress some points concerning the validity of equation (3.33). It is important to realise that in a real experiment we never measure the theoretical reflectivity as given by Eq. (3-33) since the incident beam is not necessarily strictly monochromatic, is generally divergent, and the detector has a finite acceptance. For any instrument, the effects of the divergence of the x-ray source, of the slit settings or of the angular acceptance of the monochromator and analyser crystals used to collimate the incident and scattered beams (see Chap. 7 by J.M. Gay) must be taken into account. Those effects can be described using a 3-dimensional resolution since q
--
q, in
measurable in the
--
function which is
having
a
never
certain width
a
Dirac distribution but
(see Chaps.
4 and
7)
which
a
3-dimensional function
precisely depends on the reflectivity
setup characteristics detailed above. The value of the measured For this reason, the reffectivity of a flat surface is described. which is more normally used to describe the reflection by
"Specular", a term ordinary mirror. It seems that Compton [2] was the first to have foreseen the possibility of totally, reflecting x-rays in 1923 and that Forster [3] introduced equation (3.29). Prins [4] carried out some experiments to illustrate the predictions of this equation in 1928, using an iron mirror. He also used different anode targets to study the influence of the x-ray wavelength on the absorption. Kiessig also made similar experiments in 1931 [5] using a nickel mirror. An account of the historical development of the subject can be found in the pioneering work of L.G. Parrat [6] in 1954, and of Abeles [7]. The fundamental principles are discussed in the textbook by James
[8].
as
an
Specular Reflectivity
3
Rough
from Smooth and
Surfaces
95
examples of useful data used in reflectivity analysis. The table density p, the critical wave-vector q’_ the parameter J, the absorption coefficient 0, the structure of the material and its specific mass (6 and 1.54A). A useful formula for calculating the critical wave-vector ,8 are given at A Table 3.1. A few
contains the electron
,
=
transfer is
q,(A-1)
0.0375
Si
Si02 Ge
((e-//k11)) Vp--, P__
and
conversely
2
p,
711qr
Structure
P’,
q,
e-/A’
A‘
106
107
0.7083
0.03161
7.44
1.75
0.618
0.0294
6.5
1.425
0.0448 15.05
a=5.43A,
1.317 0.0431
13.9
4.99
2330
Z=8 2200
1.7 5
P
kg/m3 cubic diamond
cubic,diamond
a=5.658A, AsGa
=
cubic,diamond
a=5.66A,
5320
Z=8 5730
Z=8
67.5%SiO2 712%
Glass Crown
0.728
0.0328
8.1
1.36
B203 9%,Na2O,
2520
9.5%K20,2%BaO Float Glass
0.726
Nb
2.212
0.03201 0.056
7.7 24.5
1.3 15.1
-
cubic,bcc
a=3.03A, CU
2.271
24.1
0.0566
5.8
Au
4.391
46.5
0.0787
49.2
Z=2
cubic, fcc
a=3.61A,
8580
8930
Z=4
cubic, fcc
19280
a=4.078A 2.760
Ag
0.0624 29.25
0.0395
28
1.08
W03
1.723
0.0493 18.25
0.334
0.0217 3.61 0.123
H20
0.32
0.0212
-COOH
0.53
0.0273
CC14
0.46
0.0254
0.268
0.0194
-
CH30H
10500
11.8
Zr02
CH3 CH2
cubic, fcc a=4.09A
12
-
1000 -
1
t-0.3-77 .0.0233
be estimated through the convolution of equation (3.33) with the resolution function of the instrument. For measurements made in the incidence plane and under specular conditions, a first effect is that the convolution can
dependence of the reflectivity. This can generally be acconvolving R(q,) with a Gaussian function. Another, most important effect of the finite resolution is that beams outside the specular direction are accepted by the detector (in other words, the specular condition J(q,)J(qv) is replaced by a function having a finite width zAq, x zAqy). Then,
smears
out the q.,
counted for by
Alain Gibaud
96
if the surface to be
analysed is rough, the convolution with the resolution drastically changes the problem because part of the diffuse intenwhich arises from the roughness is contained in the resolution volume. It sity even happen for very rough surfaces that the diffuse intensity becomes may as intense as the specular reflectivity. When this is occuring, the only way to use equation (3.33) is to subtract the diffuse part from the reflected intensity to obtain the true specular reflectivity (see Sect. 4.7 for details). function
The Transmission Coefficient
3.1.3
As shown in
Eq. (3.18),
the
I+
the
amplitude of the transmission coefficient satisfies relation, straightforward to show by combining Eqs. (3.18) and (3.29), that the transmitted intensity must be given by, r
--
t. It is
T
(0)
tt*
=
20 --
0 +
V02
-
02
-
C
2i,8
2
(3-34) 2
tt*
T
2q , q, +
The transmitted
intensity
has
(3-35)
--
VqZ
2
maximum at
2
qC
2i7r2,8 \2
O=Oc
as shown in Fig. 3.3 which intensity as a function of the incident angle 0 (or qz) in the case of silicon, germanium and copper samples irradiated with the copper K, radiation. The transmitted intensity is nearly zero at very small angles in the regime of total reflection. It increases strongly at the critical angle and finally levels off towards a limit equal to unity at large angles of incidence. The maximum in the transmission coefficient, which is also a maximum in the field at the interface is the origin of the so-called Yoneda wings which are observed in transverse off-specular scans (see Sect.
gives
a
the actual variation of the transmitted
4.3-1). The Penetration
3.1.4
The
absorption
of
Depth
beam in
medium depends on the complex part of the penetration of the beam inside the material. The refractive index for x-rays, defined in equation (3. 1) is n 6 io. The I amplitude of the electric field polarised along the y direction ((s) polarisation) and propagating inside the medium of refractive index n is given by, a
a
refractive index and limits the
--
E
Since tion
n COS
can
Otr
=
COS
be written
0
--
Eoe i(wt-kon
(the
cos
Ot,x+kon sin
-
Otrz).
Snell- Desc artes’law) and sin Otr
-
(3.36) Ot,
,
this equa-
Specular Reflectivity from Sinooth and Rough Surfaces
3
97
4
Si 3
----
2
-
E
G.
I
-
-
T F_
0
,
0.000
% ( 3.3. Transmission coefficient in
Fig. and
germanium;
0.150
0.125
0.100
0.075
0.050
0.025
0.175
A ’)
intensity in different materials, silicon,
copper,
the maximum appears at the critical wave-vector transfer of the
material
E
The
absorption
is
--
EOe+i(wt-kocosOx)eikon0trz. the real part of
governed by
nOtr
-- ::
(I
-
The coefficients A and B
6
-
can
(3.37)
eikon0trz,
-2J- 2iP i#) V’_02 -
--
with
A + iB
be deduced from the above
equation
(3.38) and B is
given by, B
(0)
v "2_
rj(02
It follows that the electric field
E
Taking
-
2J
+
4p2
-
(02
-
26),
(3-39)
is,
Eoe i(wt-ho
cos
Ox+koAz)
-koB(O)z.
(3.40)
the modulus of this electric field shows that the variation of the in-
tensity 1(z) with depth into the material is given by,
1(z) The
absorption coefficient
oc
EE*
-_
is therefore
loe- 2koB(O)z
(3.41)
Alain Gibaud
98
y
(0)
=
-2koB
and the uated
penetration depth which by 1/e is given by, zj/
(0)
(0).
In
-41rB -_
-A
I
(0)
quantity depends particular, in the limit
zi/
(3.42)
1
-
p
(0)
A
is the distance for which the beam is atten-
=
Note that this B
(0)
on
0
-
47B
(0)
21?nk-,,l
the incident
(3.43)
angle 0 through the value 0, neglecting absorption,
-+
(01)
A __
(3.44)
*
47rO,
of
addition, the penetration depth is wavelength dependent since 3 depends the wavelength. Values of 3 are tabulated in the International Tables of Crystallography, vol. IV [9] or they can also be found at the web site which has already been referred to, "http: / /www-cxro. lbl. gov/optical-const ants/". In
on
100()0
TZ Uj
100D
z
A:’ 6A c =G.0311,1 A:’
_S1
Gla, qj0.0448 Ge 1 100
Cu C’,
-
A" !I-O.OW A
Li z LU 10 0.00
0.02
O.G4
0.06
0.08
GAO
0.12
0.14
0.16
% (k)
Fig.
3.4. Evolution of the
Ka line
fi ure
is
penetration depth in Si, Ge and Cu irradiated with the of a copper tube as a function of the wave vector transfer. Note that the 47r sin O/A presented as a function of q,
Figure
3.4 shows the variation of the
penetration depth as a function of angle in silicon, germanium and copper, for the case of CuKa radiation. The penetration depth remains small, that is below about 30A when 0 is smaller than the critical angle. This is this property which is exploited in surface diffraction, where only the first few atomic layers are analysed. The penetration depth increases steeply at the critical angle and finally slowly the incident
gr(jws when 0
>>
0,.
Specular Reflectivity from Smooth and Rough Surfaces
3
X-Ray Reflectivity in Stratified Media
3.2 The
99
simple
of
case
a
uniform substrate
exhibiting
a
constant electron den-
section. This situation is of
course not previous and media stratified multilayers are frethe most general one. For example, be considered as cannot interfaces encountered. generally Moreover, quently be interfaces Thick thick. and approximated by may steps, but are rough electron of constant slabs into them as many density as necessary dividing it is not possible in their to describe (continuous) density profile. Again, the reflectivity. calculate to coefficients Fresnel the this case to use directly the be boundary conditions for The calculation must performed by applying between the slabs interfaces the each of fields at and the electric magnetic result is The electron of constant usually presented as the product density. in account in the calculation taken reflections and are of matrices, multiple excellent descriptions of Several reflection. of the known as dynamical theory
sity,
was
considered in the
this kind of calculation
3.2.1
Let
us
can
be found in references
The Matrix Method
consider
a
plane
wave
polarised
plane of incidence ((s) polarisation) The in
axes are
Fig.
[10-14].
chosen
that the
so
and
wave
is
in the direction
perpendicular
to the
propagating into a stratified medium. travelling in the xOz plane as shown
3.5.
Air 0
-
IV
+
I 2
Zj-1
Fig. 3.5. Illustration of the plane of incidence for a stratified medium. The signs an’d + label the direction of propagation of the wave; Air is labelled medium 0 and the strata are identified by I<j
59A
P: -2
()
......
-
LU --I
-3
LL
rV
-4
LU -5
-
2 U)
-6
-
> r’
reciprocity
X
large IR
located at R in as
theorem
the Green tensor is
reciprocal
symmetry relations
on
so
that fi
R/R.
gives Eq. (4-11)
in the
sense
that
the Green tensor involve
Note that the
but does not tell
9(R, r’)
:=
U(r’, R).
transpositions,
see
dipole us
that
In fact the
e.g. Ref.
[14].
Diffuse
4
sample.
IR (see Fig. 4.1).5 tangent plane
r’l
-
we
(s)
choose ’
directions
’)
X
consider
a
; (8)
X
-
a
or
e
2
(R, r’)
Green Function for
now
R
-
k.,,.r’/ko
therefore be
developed
on
:he
the
-ikoR
(4.14)
e
47rEOR
scattering, for example along ’ (P), one simply has,
normal to the direction of
(’)’(P)
We
wave can
O(ia 0
(R r)
E det
4.1.5
fi.r’
-
wave,
(p) polarisation
and
R
dipole spherical
The
Eedet If
r’l
IR
127
is observed at the
position and that the field approximation, one can develop
is here located at the detector In the far-field
Scattering
k0
the
-ikoR
(4.15)
47rcOR
Stratified Medium
planar multilayer
reference state and
as
want the
we
point r’ by a unit dipole placed in R. expression We assume that the far-field conditions are satisfied, so that the direction k"c is meaningful, and we consider the two main states of polarisation, 3 and & -_’ i(P). The point r’ can be taken anywhere in the stratified medium, the general case of a stratified see Fig. 4.2. Using the same plane-wave limit in for or (p) polarisation for r’ lying be can (s) generalised Eq. medium, (4.15) of the electric field created at
in
layer j Edet (’)’
as,
(P)
6
(R,
k
2e-
ikoR
Pw
OT7rcoR El
The far-field, conditions
(or
(8), (P)
Fraunhofer
-k,,c,,,j,
diffraction)
Z/)- (s),(p)eikc
(4.16)
SC
are more
restricting of
Indeed,
needs
r
2//\
to
neglect
to be small
quadratic compared to
R.
c
=
-
6
than
only
e-’k,)JR-r’J
expansion Applying this approximation in r’ Eq. (4.8) yields a condition on the whole size of the scattering object (since in Chap. 2 has shown that the covers all the perturbated region). The discussion in Eq. (4.8) can actually be restricted to the supp ’ort of the integral appearing domain of coherence (induced by the incident beam and detector acceptance) of the scattering processes. In this case the far-field conditions can be written as 12. h/A < R. In a typical x-ray experiment, the sample-to- detector distance is IA. The total illuminated area is a few mm but R 1m, the wavelength is the coherence length is 1coh lpm, hence the far-field approximation is valid. When the coherence length is too important (very small detector acceptance) for the far-field conditions to be satisfied, we are in the frame of the Fresnel diffraction r’l [16] [17]. and one needs to retain the quadratic terms in the expansion of IR The electric field is the solution of the inhomogeneous differential equation 2 1 2 17 x V x E’ d ,, (r’) n ref(Z )koEdet (r) ’ &S(R-r’) that satisfies out-going wave the detector position lies in medium at unitdipole boundary conditions. The 0 as depicted in Fig. 4.1. In the homogeneous region 0, the electric field can be written as the sum of a particular solution and a homogeneous solution. The parR > r’.
one
term in the
the
e
-
=
Jean Daillant and Anne Sentenac
128
(SMP) 6ik.,11.r1l
PW
E
is the field in medium i for an incident plane wave with polarisation (s) or (p) which can be computed by using standard iterative procedures [18,16]. Using the notations of chapter 3, Eqs. (3.47), (3.48), one has, i
PW
(s)’(P)
(k,,,, z)
=
T-Tj(’) (P) (k
Zj) e-"kl,j’
-
+
U(’)’(P) (-k z,3) Z3-)e’k -
(4.17) where
r
--
(rij, z)
and where the calculated for
Differential
4.1.6
It is
with
now
z
possible
first choose the
Ede,t(R,r’)
z
coordinate with the
to
Scattering
give
field scattered in the i i of
is the
superscript "PW" an incident plane
vacuum as
in
an
--
has been used to
=
wave.
Cross-Section
exact
R/R
origin taken at z Zj, emphasize that EPW is
--
expression for the ; -component of the E k,,/ko direction, E,,, Eil’. We -
-
reference state.
Eq. (4.8),
one
Substituting the expression (4.14) obtains for the component along ; of the
given in Eq. (4.13) while the general homogeneous solutions simply up-going plane waves with wavector modulus ko. In media j with 3.0 0 0 s, the electric field is solution of the homogeneous vectorial Helmholtz equation and it can be written as a sum of up-going and down-going plane waves with wavevector modulus kj. In the substrate the general solutions are downgoing plane waves with wavector modulus k,. To obtain the amplitudes of these plane waves we write the boundary conditions at each interface. The far-field approximation permits to simplify greatly the problem. In this case, the expression of the particular solution at z Z, is given by Eq. (4.15). The dipole field close to the first interface can be approximated by an "incident" plane wave with wavevector k,,,. Hence, the amplitudes of the other plane waves (that are the general solutions of the homogeneous Helmholtz equations) are calculated easily with the transfer matrix technique presented in Chap. 3. The problem has been ticular solution is are
=
reduced to the calculation of the electric field in
by
plane
a
stratified medium illuminated
superscripts (s) or (p) is always unambiguous : it indicates the direction of the radiating unit dipole in a vacuum for a given position R of the detector. In other words, it indicates the polarisation state of the scattered plane wave with wavevector k,;,. Note that the directions of ; P) , and ks,_ will vary from layer to layer due to refraction whereas the directions given by k,,11 and ; 2) do not change. a
wave.
The
.
meaning
of
Diffuse
4
Scattering
129
scattered field in direction fi:
60f
E,, (R)J6
k20 e-
dr’Jn 2(r’) edet (R)
e-ikoR ,
47rR ikoR
dr’Jn 2(r)
I dr’(k (r’)
-
41rR
r’). E (r’)
ik-.r’
ko
-
(4.18) Writing Eq. (4.18)
for i
equal respectively e-ikoR
E,;, (R)
=
scattering.
E
-
(ii.E)fi
2,
one
obtains:
ko)E-L (r’)
r
47rR
of
and
ikoR
-I dr’(k where Ej-
i,
dr’Jn2 (r) E i (r /)e
47rR e-
to
(4.19)
is the component of the field normal to the direction case the incident field.
Note that the reference field is in this
scattering cross-section we proceed by expression. In the far-field approximation,
To calculate the differential
ing
the
Poynting
vector
1il
x
deriv-
E,
C
and the
Poynting’s
vector is:
JEF
S
fi.
2yoc
scattering cross-section is obtained by calculating the flux of Poynting’s vector (power radiated) per unit solid angle in direction k ,c across of Maxwell’s a sphere of radius R for a unit incident flux. Using the linearity equations, it can also be calculated for an incident field Eil, across a unit
The differential
surface. One gets,
This exact
ko
dQ
16 7r2 I Eil 12
expression
extended detector statistical
4
du
on
properties
j
dr’6 n 2 (r’) E _L (r’) e ik,,.r’
has been used in
Chap.
the measured scattered
of
a
I
2 .
(4.20)
2 to discuss the effect of
intensity
an
in relation with the
surface.
scattering from random media, we have seen in Chap. 2 that scattering can be separated into a coherent process and an incoherent process. The latter is the usual quantity of interest in a scattering If
one
considers the issue of
Jean Daillant and Anne Sentenac
130
experiment
and it is
do-
dS2
)
given by,
fl
4
ko. 2.
1 r)
incoh
I First Born
4.2
The first Born
2
dr’Jn 2(r’)E-L
(r/)eik_.r’) 2
dr’ Sn2 (r’) E _L
Approximation
approximation which neglects multiple reflections
can
only
be used far from the critical angle for total external reflection. Close to this point, the scattering cross-sect*ions are large and the contribution to the measured
intensity of at least multiple reflections cannot be neglected. The main advantage of presenting this approximation here is that it makes the structure of the scattered intensity very transparent. It has already been presented in Chap. 2 in a different context with the aim of illustrating how statistical information about surfaces or interfaces can be obtained in a scattering experiment.
4.2.1
Expression
In the Born
evaluated in
of the Differential
Scattering
Cross-Section
approximation, both the Green function and a vacuum, Eq. (4.11).
E%t (13, d
-
r’)
ko (ii
x
e)
the electric field
e- ikoR x u
-
47rcoR
(4.22)
e
Ejje- iki_r’.
E(r’)
(4.23)
-k,,c is the
wavevector orientated from the detector to the surface which
the
field of
dipole
EW
=
EinC
are
gives
Eq. (4.10). Then, substituting into (4.11), -iki_r’
+
(Ei,,
47rR
* sc); scjdr6n 2eiq.r,
(4.24)
with the wavevector transfer: q
For such
a
field
=
k,,c
-
(4.25)
kin-
dependence, the differential scattering cross-section (power angle, per unit incident flux) is [15]:
scattered per unit solid
k4
do-
dQ
02 ( iin . sc)2
167r
Note that for small wavevector
-
j
dr Jn 2e iq.r
transfers, (’6jn.’ ,c)
1.
12
(4.26)
Diffuse
4
Example: Scattering by
4.2.2
a
Scattering
131
Surface
Single Rough
apply Eq. (4.26) to the case of a single rough surface. complicated case of a rough multilayer is treated in Appendix 4.3. This example of the diffuse scattering by a rough surface within the Born approximation is the simplest one can imagine and is mainly treated here to show how height-height correlation functions arise as average surface quantitites in the scattering cross-section. The scheme of the calculations will always be the same within the Born or distorted-wave Born approximations, whatever the kind of surface or interface roughness considered. We start from proceed
To
The
we
will first
more
ko’
dodS?
The upper medium 1) is made slightly
grating
first
over z
(n
167r 2 qz2
dQ
Equation (4.28) k 04(n 2
1)2
2
167r qz2
dQ
vacuum)
or
_
dz
is medium
j drjj
_)2
e
iq.r
00
0, and
in order to make the
1)2
2
(r[l)
dril
2
(4.27)
the substrate
integrals
(medium
converge. Inte-
2 e
iq,, z (r1j)
iqjj.rjj
(4.28)
be written:
can
_
_
(air
I f
1)2 ( in -_; Sc )2
2
absorbing yields,
k40
do-
do-
z
(n 167r2
-
(; in sc) -
2
dr’ll e iq,, (z(rjj)-z(r’jj)) e iqjj.rjj -iqjj.ir1jj
drjj
(4.29) Making
the
change
do-
k 4A 0
dQ
167r 2 qz2
of variables
(n
2 _
R11
=
ril
-
1)2 (; in .-6sc )2
r’ll
and
integrating
over
R11:
dR11 (,iq.,(z(Rjj)-z(O))),iqjj.Rjj (4.30)
where A is the illuminated
area
and
we
have
simply
used the definition of
surface.’ Assuming Gaussian statictics of the height flucChap. 2), or in any case expanding the exponential to the order, we have:
the average over a tuations z(rjj) (see
lowest
(second)
(eiqz(z(Rjj)-z(O)))
-
--Lq.’,(z(Rjj)-z(O))’.
(4-31)
We then obtain: do-
k 40 A -
dQ
167r2q2
(n 2
2
-
1) (; in. &Sc)2
e
_,7.2
2
dR11 eq2 (z (R11)z
(0)) eiqll. R11
z
(4.32) 7
In
general,
this average over the surface will not be known and as discussed in Chap. 2.
ensemble average
we
will
use
an
Jean Daillant and Anne Sentenac
132
This
equation also includes specular (coherent) components because it has general solution of an electromagnetic field in vacuum. The diffuse intensity can be obtained by removing the specular
been constructed from the a
component:
8
k4 A
(dfl)c.h do-
where the
identity
"0"
4q2
has been used. The diffuse
dQ
_
choose
now as
interfaces
k4A 0
-
-
167r 2 qz2
incoh
present
an
(n
2 _
index
(qjj)
--
1)2 ( iin .,; sc)
(4.33)
(4-34)
is then:
2
(Z2)I dR11 (eq",:(z(Rjj)z(O)) 1)6iqjj.Rjj _
a
further order of
complexity.
We
the real one, but with smooth The Green function and the field in Eq. (4.11)
same
profiles).
(4.35)
Approximation
approximation with
reference state the
(step
. sc)2 S(qjj),
(incoherent) intensity
Distorted-Wave Born
We will
q 2 (Z) 2
e
j dR11 eiqjj.Rjj
xe-q.2
4.3
1)2
for Dirac S functions I
)
2
z
47r2
do-
(n
system
as
therefore those for smooth steep interfaces and the iterative methods discussed in Chap. 3 can be used to calculate the field and the Green function. are
This
approximation yields better results than the first Born approximation angle for total external reflection. It is currently the most popular approximation for the treatment of x-ray surface scattering data. A first change due to the new choice of reference state is that, because near
the critical
refraction is taken into account, the normal component of the wavevector now depends on the local index. Using Snell-Descartes laws:
k,,j
=
koVSin2 0
-
sin
2
Oc,’
(4.36)
Integrating (4.33)’ over the angular acceptance of the detector So SSdetector/R 2= dOdo (2/koq_,)dqjj, and normalising to the total incident flux through the area A (leading to a factor A sin 0, since contrary to the reflectivity coefficient daldfl is normAlised to a unit incident flux), one obtains for the
=
=
reflectivity coefficient R: 4
R
=
OA (n2 4
qz
)2
e
-q2(_)2
(- in -’ s ) c
2q,
2q.,
e
_q2(z)2 z
(- &i. I SC)2
which shows the well-known qz 4 decay. This expression also shows that within the Born approximation, the Brewster angle is 45 degrees.
Diffuse
4
where
OC,
vacuum
=
2(1
ni)
-
is the critical
and medium i with ni
imaginary
=
angle I
-
Ji
I -
v’2
Irn(k,,j)
ko
[(02
I- ko V[(02
-
i#j.
-
More
+
4#j2j1/2
2Si)2
+
40i’]1/2
-
precisely,
the real and
are:
2Ji)2
V2
-
133
for total external reflection between
parts of the wavevector in medium i
Re(k,,j)
Scattering
+
-
(02
(02
-
-
2Jj),
(4.37)
2Jj).
(4.38)
above, refraction also implies that the direction of the polarisation vector in (p) polarisation changes from layer to layer. To avoid the complications related to this point, unless otherwise specified, we will always limit ourselves to the case of scattering of a (s) polarised wave into (s) pocos 0, in every larisation in the rest of this section. Then, one has (; in -; sc) will be effects of given in secpolarisation layer. A more detailed discussion As mentioned
=
tion 4.4.
4.3.1
Case of
a
Single Rough
Surface
Considering only one rough interface between media (0) and (1) and placing the reference plane above the real rough interface (Fig. 4.1),’ we have, for 9
The choice of the reference erence
planes
is
important
medium, here. This
particular of the location of the refquestion did not arise in the discussion of
and in
approximation where the reference medium is the vacuum. Three different choices are a priori possible: place the reference plane above, below, or crossing the real rough interface (Fig. 4.2). In principle, all choices are equivalent for small roughnesses owing to the continuity of the field. For larger roughnesses, using the average plane of the rough surface might be the best choice. This has been done to calculate the specular reflectivity in Ref. [1] close to the critical angle. However, this approximation is not good for larger incident angles because Fresnel eigenstates are not a good approximation of’the real eigenstates of the system. In this regime, however, the first Born approximation is good far from Bragg peaks. In the present treatment, we have choosen to place the reference plane above the rough surface, hence the e. 1 and t’0 coefficients. This approximation is as good as that of the average plane close to the critical angle and of incidence. For a converges to the first Born approximation at larger angles multilayer, one might be worried by the phase factor corresponding to the small shift between the average plane, and the reference plane placed below the surface (this phase shift disappears in the cross-section for a single interface). On the other hand, when using the average plane, calculations become cumbersome. A reasonable solution is then to chose the average plane as reference plane, but to that use the analytical continuation of the field in one of the media, for example the first Born
n ,
above the interface.
C
Jean Daillant and Anne Sentenac
134
(s)
or
(p) polarisation: Ee
,c
det
k2e-ikOR 0
(R, r’)
4ri coR 2
k oe
-ikoR _
-
4rcoR E
eikscj.r’ &Sc,
tscl 0,
(4.39)
,
EjPW (kin z, 13 /) 6- ikjj.r’jj ; in
Ein =
Epw(-k,,,,l,z)e’’k 11-r1les, 1
-
z
in
Ein
toj
e- ikj.,j.r’ ’6i.
(4.40)
1
where t" and tsc
are the Fresnel transmission coefficients for polarisation (8) respectively the angle of incidence Oin and the scattering angle in the scattering plane Os,. Explicit expressions for those coefficients are given by equations (3.71) and (3.72). Putting Eqs. (4.39), (4.40) in Eq. (4.11) and following the same treatment of the integrals as in Sect. 4.2.2, we obtain a generalisation of Eq. (4.35):
for
(dQ)incoh do-
A
- L (n 167r2
X Cos
20
-
2 1
-
n
2)2 0
Itin,12 Itsc,12 0, 0,
e
.L(q2’j+q*2’1 (Z2) 2
-
iq,,112
1 dRjj 16 jq,,1j2(z(Rjj)z(O)) I Ciqjj.Rjj _
I
.
(4.41)
Eq. (4.41) differs from Eq. (4.35) by the additional transmission coefficients. This
expression is explicitly symmetrical in the source and detector posirequired by the reciprocity theorem. At the critical angle for total external reflection Oin 0, the transmission coefficients in Eq. (4.41) have a peak value of 2. The electric field is then at tions
as
--
its maximum value at the interface because the incident and scattered field 0. As the dipole source equivalent to roughness Co6n 2 E phase at z is proportional to E, there is a maximum in the scattered intensity. By using the reciprocity theorem, one can see that the Green function is also peaked near Osc 0,.’o Those peaks are the so-called Yoneda peaks [19]. They can be seen on Fig. 4.4. are
in
--
=
4.3.2
General Case of
In the
general
layer j
for
(s)
case
or
of
a
Stratified Medium
stratified medium
depicted in Fig. 4.2,
one
has
in
(p) polarisation: k 02e- ikoR
E’dlelt (R, E (r)
a
47rEoR "Zz
Efw(-k,,,,, ’i z’)e ik_ 11.r’jj-, s,:j 3
W Ein Ef (k-in z,31 3 .
Z/)e-ikj,,jj.r’jj-i ,nj.
Equivalently, the peak in the Green function can be dependence of the field emitted by a dipole placed
seen
to arise from the
below the interface.
(4.42) angular
4
Diffuse
Scattering
135
no
zi n
Z2
nj_j
zj (X,Y)
Z.
zi
nj j+1 n
N
ZN+1 Fig. 4.2. X-ray surface scattering in a stratified rough medium. Because of multiple reflections, there are waves propagating upwards (with an amplitude U(k,,j, z)) and downwards (with an amplitude U(-k;,,j, z)) in layer 3 where the total the field amplitude is EZ (there is an equivalent dependence of the Green function). in,3 Multiple reflections are considered within the DWBA but not within the first Born approximation. The perturbation method consists in evaluating the field scattered by the dipolar density equivalent to the index difference (nj 1 nj) between the real system where the rough interface profile is zj(rll) and the unperturbated system where the interface is located at Zj, and is placed here at the average interface plane. For interface J* the unperturbated and real index distributions differ in the hatched region -
-
The DWBA method consists then in in
Eq. (4.17) PW
Ej,
in each medium as, for
(’)
(k
,
31
Z)
U (’)
developing the EPW functions defined example in (s) polarisation, in layer j:
(k ,j) Zj )
U
ik; ,
e -j’
+
U()(-k ,, zj)e+ik ,jz ,
k ,j, Zj)jik ,jz,
(4.43)
magnitudes of the upwards and downwards are explicitly obtained in Chap. 3 of this book, Eq. propagating waves Vincent" representation of tranfer matrices [20]. and (3.48) using the "Vidal
where the U coefficients
are
the
which
The field is then written
(put Eqs. (4.42)
in
Eq. (4. 11) and
sum
over
all
Jean Daillant and Anne Sentenac
136
interfaces):
E(’)
--
E,, f + Ej "
J
(r1j)
z j+j
k2e-ikoR 0
dril eiqlIxII
47rEOR
j=O
dzE-O(n +j
-
3
0
1;,PW n2) i _j+I (ki,,,,,j+,, z) -’,’Pw(-ks,, 3+1
Z),
(4.44) where it has been assumed that the reference
interface, hence the EPPW 3+1 fields, zj+l of Eq. (4.41) is: dodS2
)
N
N
j=1
k=1
k4 ’0 DT7r2
incoh
plane is located above the negative). Then, the generalisation
is
U(’)(ki,,.,,j, Zj)U(’)(k , ,j, Zj)U(’)*(kj,, ,,j,, Zk)U(’)*(k,,,_,,k, Zk) Qj,k (kin zj
k , zj, kin z,k
k.,c z,k),
(4.45) with
(n
Qj,k (qz, qz)
3
n _ 1) (n 3
0
2
k
n
2
k
_1)* COS2
dril
j dr’l
1e
iqjj.(rjj-ir’jj)
(fzj(rll)
0
0
dz’e i (q;, z q,,,* z’)
dz
-
( fk(r’jj)
dze’.q,,
z
0
dz’e- iq.,"zl
(4.46) specular (coherent) contribution, obtained as an average over the Chap. 2, has been removed. Performing the integrations over z’ and making the change of variables r’ll RII as previously: r1l
where the field z
as
and
shown in
-
Qj,k(q ,, q’) z
=
A-
W3-
-
0 3
-
1) (nh2
_
n2_j* h
qz qz
dRii
e
iqjj.Rjj
COS
2
Oe- [qz2 (Z+ 2
2
(Z2)] k
(eq.,qz’*(zj(O)zj,(Rjj)) (4.47)
Because reflection at all interfaces is taken into
account, all the possible
com-
binations of the incident and scattered wavevectors appear in the formulae.
4
Diffuse
Scattering
137
100
.911
on a liquid surface as a function reflectivity curve. Note that in the case represcattering is peaked in the specular direction. The roughness spectrum (Fourier transform of the height-height correlation function) is directly obtained in a q., scan at constant q,, The possible extension of q., scans increases
Fig. 4.3. Intensity
scattered
of q., et q-.. I(q,, q , = sented here, the diffuse
0)
by
a
2nm thick film
is is the
with q-.. Out of this accessible range (black surface on the figure), either the detector or the source would move below the interface. Note the Yoneda peaks near the
extremety of q., scans, where 0j.
or
0,, is equal
to the very small critical
angle
for
total external reflection
Particular Case of
4.3.3
a
Film
particular case of a film has been considered in Ref. [2] where an explicit expression for the scattering cross-section as a function of the reflection and transmission coefficients has been given. In practice however, it is more convenient to write a program using Eq. (4.45) whatever the number of layers. In a rough thin film scattering occurs at both the film surface and at the film-substrate interface. The field at the film surface is proportional to 1 + r where r is the film reflection coefficient, and the field at the substrate-film The
interface is
depend
on
proportional to the film transmission coefficient 0ij et 0,,,, Eq. (3.75), (3.76): 2ik ,,jd roj + rl,2e2ik, +
r0,17’1,2e_
where rj,j+j ti,i+l i1i + 1, d the film Far from
an
reflection,
Then,
the Born
Green
function)
Therefore,
3.
r
and. t
d
(4.48)
+ rojrj,2e -2ik.,,ld
the reflection et transmission coefficients of interface
thickness,
incident
ternal
face.
are
10,1t1,2e_ ik.,,
; t
t. Both
or
and
exit
k,,, depends
angle
the reflection coefficients
approximation
is
on
0j,,
or
0,
close to the critical
valid,
are
angle
small and
and the
can
amplitude
for total be
ex-
neglected.
of the field
(or
is I at the upper interface and e- ik,,,ld at the lower interneglecting the polarisation factor, the scattering cross-section
138
Jean Daillant and Anne Sentenac:
will be
(compare
Eq. (4.C3)
also to
4
do-
ko 167r2q2
dQ
2
+(,17,2 Following
the usual
( dQ)incoh [(n,
appendix 4.C):
j drjj [(n
z,
2 1
n
_
ei(ksc, -k1nz)Z1 2) 0
nfle one
obtains-
I
(compare
A
2)2
-no
2
+(n2
e
-qf (Z:2L
dR11
(e q,’,(zi.(O)zj(R[j)) 1)6 jqjj.Rjj
dR11
(eq
2)2 e q2(,2) -
2
.
-ni
2
+2(n2
-
2) (n,2
nj
_
2
(Z2 (O)Z2(Rjj))
2)r- Lq 2(Z2)
-no
2
-
2
_
_
j dR11 (e qz2(zj(O)Z2(Rjj)) principle,
Eq. (4.CIO)):
to
167r q.
2
In
(4.49)
iqjj.rjj2
i(k,,: ,-kjz)(d+z2)j
2
-
procedure,
do-
in
_,
1)
lqz2(Z2) 2
f 2
z
_
e
jqjj.Rj1
(4.50)
cos(qd)
1) Cjqjj.RjjI
the correlation between the different interfaces
can
therefore be
determined because the contrast of the interference pattern directly depends on this correlation and the different contributions may be separated [21].
substrate, the substrate rougha film. Then, qz scans at conspectrum stant q, can be performed with the film. They consist of an oscillating and a non-oscillating component. The contrast of the interference pattern yields For
example, considering can
ness
single
a
film
on a
be first measured without
the cross-correlation between the film-substrate and film-vacuum interfaces at the
given
q, and. the
non-oscillating part yields
the
sum
of the film-substrate
and film-vacuum auto correlations. All the relevant correlation functions
therefore be determined. In
a
multilayer,
can
similar constructive interference ef-
interfaces, in Sect. 8.5. scattering" leading In addition to the Yoneda peaks, other dynamical effects can be observed in the case of a film (or better of a multilayer, see Sect. 8.5 for a thorough discussion). Because the reflection and transmission coefficients Eq. (4.48) depend on both the incident and scattering angle if the reflection coefficients are not too small, the scattered intensity can show oscillations with a characteristic period depending on the film thickness,11 even if both angles vary in a scan so that qz is kept constant (Fig. 4.4 left). This dynamical effect cannot be accounted for within the first Born approximation. More generally, similar dynamical effects will occur whenever the field is modified at an interface due
fects
can occur
between the beams scattered at different conformal
to what is called "resonant
Note that because of thee -2ik;,
associated to the cross-correlation
dfactor in
the reflection
coefficient,
the
periodicity
effect is different from that associated to the interface
dynamical term, allowing
to
distinguish
between them.
Diffuse
4
multiple reflections (see Fig.
to
right
4.4
for
are
not accounted for within the first Born
are
given
Scattering
139
example). Again such effects approximation. Many examples an
in Sect. 8.5.
10-1
10-1 10-1 5 2
10-7 5
10-6 0
8
.P
10-1
128 0 5
0
120-
0 0
.10-1
z,
-
I 0-
10-1
10-1
-
-
d-
0
o
10-5
o
10-1
1
%
10
1.0
0.5
0
-0.5
1.0
-6
-0.8
(X10’ m-)
Fig.4.4. Dynamical
0.4
0
0.8
(X 101)
effects.
Left:
"Rocking
(polystyrene-polymethylmetacrylate film is 18.9nm thick. Note in
-0.4
diblock
particular
curve"
copolymer)
the Yoneda
peak
its structure is related to interferences in the film
I(q,,) on a
for
at 0.75 X
(inset).
In
a
polymer
film
silicon substrate. The
10-8 m-1
particular,
’and that the oscil-
by arrows correspond to the dynamical effect discussed in the text. Right: "Rocking curve" I(q.,) for a CdTe multilayer (20 layers). The height-height
lations marked
correlation function is
(z(O)z(x))
=
a
2
exp
-
[Xg]2v
and the interfaces
are
assumed
0.25nm, fully correlated. The parameters used in the calculation are a v 300pm. The grey curve (divided by a factor of 2 for clarity) corre0.6, sponds to the first Born approximation and the black curve to the DWBA described in the text. Note the peaks in the DWBA intensity which are a dynamical effect to be =
and
=
=
occur
when the field is maximum at the different interfaces
Jean Daillant and Anne Sentenac
140
Polarisation Effects
4.4
According
to
Fig. 2.3,
the
polarisation
vectors
0
_’ ii(n
are:
sin
Oin
Cos
oin
in
0 -
sin
-
(P)
Cos
SC
-
SC
sin Os,
cos
(4.51)
0,,, sin Cos OSC
sin
0
-e(’) and scattering angles ’ is(c) in and almost are equal. (p) polarisation (s) reflection coefficients for the perpendicular a single interface:
At very small incident and and the EPW functions for
example, the ratio of the parallel polarisations is for
SC
-_
’r(P) where
00 and 01
are
with the surface. In
I +
angles between the incident and the general, at least one of the angles Oin
(p)
and
refracted beam Or
Os, is small,
reflection coefficients is also
generally not very important, but they Firstly, subtle effects can happen under conditions close to those responsible for the structure of the Yoneda peaks discussed above [4]. More importantly, polarisation effects must be taken into 10 account whenever the incident or the scattering angle are larger than degrees. This is the case when one tries to get information at small (atomic or molecular) lengthscales. The treatment of polarisatio’n effects will be different within the Born and distorted wave Born approximations. Within the Born approximation (see Sect. 4.2), the polarisation dependence is easily included in the differential scattering cross-section Eq. (4.26) small. can
Thus, polarisation effects
(s)
and
(4.52)
20o0i,
the,
and thus the difference between the
in
For
be noticeable in
some
are
instances.
du
k4
dS?
167r
2 .
2
02
dr
n 2 e iq.r
only dependence is through the scalar product ( in -’ 3sc). There is only simple, geometrical depolarisation corresponding to the projection of the incident polarisation on the final one. Generally, the polarisation of the scattered beam will not be known a priori, but we can calculate the relative scattering into (s) and (p) polarisations using Eq. (4.26). Then, since the a
dodQ
)
do- () _
tot
dQ
do- (P) +
dQ
*
(4.53)
4
Diffuse
Scattering
141
DWBA, we must always decompose the incident and scattered into (s) and (p) polarisations because the EPW functions depend on the polarisation. Moreover, for (p) polarisation the orientation of ; i,, ahd’ ., will differ from layer to layer because of refraction. Taking these two requirements into account, the scattering cross-section can be calculated using Eq. (4.53). A simple case" is that of the scattering of a (s) polarised incident wave into a (s) polarised wave since a unique cos’ 0 polarisation factor can be used in all the layers. This is the case which was considered in Sect. 4.3 for simplicity. Within the
field
Scattering by Density Inhomogeneities
4.5
Only surface scattering has been considered up to this point. However, the inhomogeneities leading to scattering can also be density fluctuations. This should always be born in mind when interpreting experiments. The scattering due to density inhomogeneities in a multilayer or at a liquid surface can be treated using a formalism similar to that used for surface scattering. The relevant correlation functions will be of the form (Jp(O, z’)Jp(rjj, z)). This problem was considered in the early paper of Bindell and Wainfan [22]. Again we limit the discussion to the scattering of a (s) polarised incident wave into a (s) polarised wave. dielectric index
4.5.1
Density Inhornogeneities in
a
Multilayer
are assumed to be perfectly smooth in this analysis. Within 13 DWBA, and assuming effective U functions within the layers, the differential scattering cross-section will be (cf Eq. (4.45)):
The interfaces the
4
do-
ko _
dS2
167r 2
N
N
J:EEj:ET U(’)(ki ,,,j,Zj)U()(k,;c ,,j,Zj) j=1
k=1
U(’)*(kinz,k)Zk)U(’)*(k,,c 12
13
Zk)j3Jj, k ( kinzj k,c ,j, ki,,z, 1,
k) (4-54)
kcz
example the case of horizontal scattering on a horizontal surface at a synchrotron source. Assuming effective U functions within the layers is only possible if the characteristic size of the inhomogeneities is much smaller than the extinction length, see appendix IA to Chap. 3. This might not be the case for multilayer gratings (see Sect. 8.7) or large copolymer domains [23]. This is for
Jean Daillant and Anne Sentenac
142
where
now
Bj,k (q,, q ,,)
-_
cos
2
drjj
j dr’jje
Z
zj+l-zj
JO
qll -(’11"10
I-Zk
dz’
dz 0
(6nj2’(rjj, z)Snk* (r’ll,
z
C
i(q,,z-q,,,*z’).
(4.55) Making
of variables r1l
change
the
Bj,k (qz, q,,)
A
--
cos
2o
I dR11 (
c
In the
case
U, (- k,
z,
of
a
semi-infinite
1, Zj)
=
Vc
are
-
r’ll
iqjj.Rjj
-4
JO
R11: Z
zj+’-Zj
1-Z
dz’
dz
(4.56)
0
Jn? (0, z) Jn k2* (1111, z’) j
e
i(q,,z q,,,*z’) .
t" and medium, only, U, (-kin, z, 1, Zj) Re Writing qz, 1 (qz, 1) +ilm (qz, 1), --
different from 0.
=
obtains:
one
B1, 1 (qz, 1,
qz, 1)
-_
A
cos
2o
I dR11
e
iqjj.Rjj
0
0
f foo
dzdz’
e
i7Ze(q ,,.)(z-z’)Clm(q.,,,)(z+z’)
Jn 2(0’
z)Jn
2*
(1111, Z’)
.
(4.57) i.e.,
the bulk fluctuations
are
integrated
over
the
penetration length
of the
beam.
Comparing Eq. (4.57) to Eq. (4-47), we note that contrary to bulk scattering, surface scattering is inversely proportional to the square of wavevector transfers. Therefore, surface scattering will generally be dominant at grazing angles whereas bulk scattering will ultimately dominate at large scattering angles (see Chap. 9, Sect. 9.3.1 for an example). Density Fluctuations
4.5.2
at
a
Liquid
Surface
interesting case is that of density fluctuations at a single liquid surface because an analytical calculation can be made. The liquid extends from -oo to 0 in z, and its vapor can be taken with negligible density. Eqs. (4.54),(4.57) above give: An
k
do-
dQ
167r
0
0
4
_"O
A(I 2
-
n
COS2 0 2) 2 Itinj 0, 12 0, 12 Itscl
f. f
dz’
dz
e
iq,,,Iz e- i q,,*,, z’
00
(Jp(O, z’)Jp(Rjj, z)) iqjj.Rjj e dR11 P2 (4.58)
4
where
we
have used
n
-
lation function for bulk
I
-
(A 2/21r)r,p. Inserting
liquid
-
-1/V((9V/OP)T do-
4 k 110
dfl
167r2
Further
4.6
A(I
the
Scattering
density-density
143
corre-
fluctuations
(p(r)p(r’)) where NT
Diffuse
=
P2 kBT KT6(r
is the isothermal
-
n
2)2
Itin12 ItscI2
-
r’),
(4.59)
compressibility yields, kBTr-T Cos
21m
2o.
(4.60)
Approximations
always intensity close to the critical angle for total external reflection [24]. Understanding scattering at grazing angles is highly desirable because bulk scattering is minimized under such conditions. This is critical because the signal scattered by surfaces or interfaces is generally very low. Differents approaches have been attempted to improve the DWBA. Only the first one has been extensively investigated. This approximation consists in taking into account the average interface profile in Eq. (4.11) [4]. The reference medium is now defined by the relative I/A f c(q) z) drjj in the case of a permittivity Cref W (6) W with (c) (z) random defined surface an by ergodic process. (In [3] the shape of the rough is an hyperbolic by tangent profile to simapproximated average permittivity main interest of this The Green the of reference the calculation tensor). plify
approximation enough representation
presented
here does not
The Distorted-wave Born
as
allow
of the scattered
an
accurate
--
new
--
reference medium is that the reference field Eref is that of the transition
directly the N6vot-Croce factor in the reflection co0 and perturbation 6n’(rll) is of null average (6n 2) we may expect to have minimised its value (and thus extended the validity domain of the perturbative development). This improvement has been shown to Yield much better results than the classical DWBA in the optical domain where the permittivity contrasts are important [25]. In the x-ray domain its interest is more questionable since it does not lead to simple expressions for the scattering cross-section. Indeed 6n 2(r1j) is no longer a step function and the integration along the z-axis cannot be done analytically. Another possibility would be to directly take into account multiple surface (roughness) scattering without using the effective medium approximation. It is then necessary to iterate the fundamental equation (4.11)[4], [26-28]. This has been done up to the second order in Ref. [5] for specular reflectivity, and the corrections might be important close to the critical angle for total layer
and thus. contains
efficient. Moreover the
=
external reflection.
Finally there exist many approximate methods that have been developed totally different contexts (optics, radar). In most methods, the field scattered by the rough surface is evaluated with a surface integral equation (given by the Huygens-Fresnel principle (or Kirchhoff integral) [29]). The integrant
in
144
Jean Daillant and Anne Sentenac
of the latter
contains
the field values and its normal derivatives at each
point approximation consists in replacing the field on the surface by the field that, would exist if the surface is locally assimilated to its tangent plane. This technique, when applied to the coherent field yields the famous Debye-Waller factor on the reflection coefficient. It is a single scattering approximation ( also called Physical Optics approximation). The perturbative theory (the small parameter is the rms height of the surface) has also been widely used. A possible starting point is writing the boundary conditions on the field and its derivative at the interface under the Rayleigh hypothesis. A brief survey of this method is given in Sect. 3.A.1. Note that the iteration of these methods permit to account for some multiple scattering effects, but the increasing complexity of the calculation limits their interest. It is now also possible to consider the resolution of the surface integral equation satisfied by the field without any approximation (and thus to account for all the multiple scattering). Preliminary results have been already presented in the Radar and optical domain. However in the x-ray domain those techniques have a major drawback: They only consider surface scattering (with a surface integral equation) and-the generalisation to both surface and volume scattering is not straightforward. The differential method [30] which consists in solving the inhomogeneous differential equation satisfied by the Fourier component of the field (in the k1l space) with a Runge-Kutta algorithm along the z-axis would be more adequate. It has already been used to calculate the diffraction by multilayer gratings and accounts for all multiple scattering (no approximation), but it remains difficult to use it for non-periodic (rough) surfaces because of the computing time and memory required. of the surface. The Kirchhoff
4.7 4.7.1
The Scattered
Expression
Intensity
of the Scattered
Intensity
specular reflection where the specular condition J(q,,,) implies that resolution effects amount to a simple convolution, the scattered intensity is proportional to the resolution volume for diffuse (incoherent) scattering. In order to achieve quantitative information from an experiment, it is necessary to measure (and calculate) absolute intensities, and therefore to have a detailed knowledge of the resolution function. The diffeFential scattering cross-section must be integrated over the detector solid angle Qd and the incident angle angular spread G AOin in the vertical and ’ Oy perpendicular to the incidence plane) (Fig. 4.5). Assuming a beam cross-section 1, x ly and a Gaussian angular distribution of the incident beam intensity: In contrast with
1C)e of width
incidence
so?
6 02
2.AO2
2AO2
i.
Y
zA0i,, in the plane of incidence and AOY normal to the plane of (Fig. 4.5), the scattered intensity ID is, using the definition of the
Diffuse
4
cross-section
scaftering
flux in direction
1D /-10
I
dMi. dS OY
-
X
unit solid
angle for
145
-
unit incident
a
ki,,):
-
IX
(power radiated per
Scattering
ly
27r, AOil,
X
’ Aoy
e
Be?
5,92
24,9? 2 19i.
2A02Y
do
Y
dS?
dQd.
(4.61) The normalisation factor
defined for
a
1/(I.,
x
1y)
is
required
because the cross-section is
unit.incident flux whereas the scattered intensity is normalized
to the total flux.
Since each experimental setup is different, it is impossible to give expression for the angular dependence of the resolution function. examples can be found in Chaps. 7, 8 and 9.
general
Different
Incident beam
Detector
AQ
a
d
OuridUe
Fig. 4.5.
Definition of the
angles
and solid
angles
used for
calculating
the scattered
intensity
The cross-section is defined
might the
be found
use
of
a
more
as a
convenient to
function of wave use wave
vectors;
and sometimes it
vectors instead of
wave-vector resolution function is very delicate
angles.
In
fact,
(see Ref.[31])
14
absolutely needs an analytical expresview, a numerical integration of Eq. (4.61) which very generally reduces to a multiplication with the detector solid angle is much more preferable. For this reason, we only give here a brief account of and should be avoided except if a computing point of
one
sion. From
how wave-vector resolution functions
The major into
a
problem
can
be dealt with.
is that the transformation of the
wave-vector resolution function leads to
separable
in q, and q,,
(hence
the
projection
in
a
angular
resolution function
function which is
Fig. 4.6).
generally
not
146
.
4.7.2
Jean Daillant and Anne Sentenac
Wave-Vector Resolution Function
Since a rigourous transformation of the angular resolution, function into a separable wave-vector transfer resolution function cannot be made generally, we use here a more simple approximation where only the resolution volume is conserved in the transformation, see Fig. 4.6. According to Fig. 4.6, close to the specular, a factor
60j" ’ Ao"
2 X
) o qZO
n
Osc’ Oin)] (and
must be introduced for resolution volume
ables, whereas there is no factor 2 if for scattering close to’the specular, an
diffuse
(in general zAO,
ly V2_7r, Aojn J
I
-TD / 10
conserva-
angular to wave-vector variexample Oin < 0,,. Considering approximation of Eq. (4.61) is: do-
2zAOi,,
I
1,
x
60in)- R(6q)
(4.62)
d6q R(6q)
-
-
>
intensity)
therefore
the transformation from
performing
tion when
k2oq,AO,, dQ
is the resolution function in the wave-vector
space.
Considering
for
1/10
k3 o =
the
simplicity
case
of
a
single rough interface,
(4.41), integrals
compute instead of Eq.
-
2)2
2
7r2 (7z Sin Oin
(n,.-
no
ItscJ12 Itin112 01 0,
e
--1(q2,j+q,,*211 )(z2) 2
I dJqll I dR11 [ejq ,jj2(z(Rjj)z(O)) 11 V2_7rko, AO,
integration since the
over
Jqz
integrant
resolution function is therefore:
tegration
over
6%,
yields
fZ,
-
R(R11) where
Aq_-NI -Log2
and
--
has been does not
7Z(Jqll)
=
C os
jqz, 112
_
where the
must
we
now
of the form:15
2
0
(4.63)
R(Sqjj)e i(qjj+6qjj).Rjj
replaced by
significantly e- 1/2
a
factor
vary
over
V2_7r, A qz Aqz. The
Jq,2 /Aq2-1/2 &qylAq 2Y. The in-
the Fourier transform of R:
27r,6_q.,zAqye_ 1/2(,dq2X2+,Aq2Y2) X
/AqyV/2_Log2
are
Y
(4.64)
the half-width-at-half-maximum of
According to Eq. (4.64), 11zAqx and 1/zAqy represent the coherence lengths along x and y, i.e. the lengths over which correla16 tions in the surface roughness can be observed, as discussed in Chap. 2. the resolution function R.
equation also shows that it is important to correct for the illuminated area, Chap. 7 and in particular Sect. 7. L In the case considered here, normalisation to the incident intensity instead of the incident flux in the differential scattering cross-section leads to the I/ sin Oi. factor in Eq. (4.63). For a typical experiment where the resolution is mainly determined by a slit of 0. Imm) x (hy H x V size (h,, lomm) placed in front of the detector, Im away
This see
16
=
=
Diffuse
4
Scattering
147
(b)
(a) Aq,
Aq,, T Z
Aq,
Aq,
AO
in
AO X
Fig. 4.6. Resolution surfaces in the plane of incidence (thick parallelograms) and their projections onto x and z in the wave-vector transfer space (grey rectangles) for scans in the plane of incidence. The parallelograms are obtained by convoluting the incident angular spread by the detector angular spread. With appropriate scaling (multiplication by ko), all points within the parallelogram correspond to possible final wave vectors, and the wave-vector transfer can be obtained through a -kin translation. Due to (x, z) coupling, the resolution area cannot be simply expressed as a function of Aq, and .6q__ In particular, because of the projection onto x Aqx x Aqz $ koq ,, AOi.AO, where koq, or z in the definition of Aqx and Aqz, is the Jacobian of the transformation. The approximation used here consists in using Aqx, Aqz, with an appropriate factor depending on the geometry for area (i.e. intensity) conservation. (a) Reflectivity geometry. The area of the parallelogram is 1/2ko q,, max (,60in lAOs,, 60 /,Min). (b) grazing incidence geometry. The area of the parallelogram is k 02q_ max(,!AOin/,AOsc1 AOsc/ZAOin)- In this last case, there is no coupling, and the transformation can be done via a straightforward Jacobian transformation. For scans limited to the plane of incidence as here, the slit openings normal to this plane only introduce an additional koA, in the resolution volume
intensity scattered by finally be written: The
a
interface close to the
single rough
specular
can
10mrad and a wavelength sample, for a typical scattering angle O , 10-2 rad in the horizontal O.Inm, the acceptance of the detector is ’AO’ 10-4 rad in the vertical. The coherence lengths along x, y and z are and AO-. lpm. This is 0.01pm, and A/AO,, 100mm, A/, AO, respectively A/O,,AO,
from the A
=
=
=
=
along of height
which allows the measurement
or
lengthscales.
incidence geometry correlations over such a large
due to the
x
range of
=
=
=
the enhancement in the coherence
density
grazing
148
Jean Daillant and Anne Sentenac
3
1/10
k0 2
87r q_, X
-
2
-(n 1 sin On,
I dR11 [e
In the limit of small qz
k0
1
10
81r 2 qz sin Oin
(n,
2)2 Itin, 0, 12 Itscl 0, 1
2 e
-
develop
one can
I
1 q ,, 1 12
)(Z2) Cos
2
116iq11.R11 ’&(R11).
the
v)
(4.65)
exponential in Eq. (4.65),
’in COS2 2)2 1 to, 1 12 1 tscl 0, 12
no
-
1(q2 2 -, +q,2
n0
lq,,, 12 (z(R11)z(0))
2
-
-
"
z(%)z(-q11)) (D R(qll). (4.66)
The scattered
roughness 4.8
intensity
is then
simply proportional function.
to
a
convolution
of
the
spectrum with the resolution
Reflectivity
Revisited
2 intensity decreases as qz for small q,, values, whereas it was shown in Chap. 3 that the specular (coherent) 4 intensity decreases as q,- One therefore expects that diffuse scattering will eventually dominate over the specular reflectivity. Of course the wave vector at which diffuse scattering becomes dominant will depend on the experimental resolution since the diffuse intensity is proportional to the resolution volume. In fact, for reasonable experimental conditions, the corresponding wave vectors are rather small, on the order of a few nm-1, and this leads to major difficulties in the treatment of reflectivity data. A "reflectivity curve" -I(q,,) is indeed never a pure specular reflectivity curve. Moreover, the diffuse intensity is often (but not always) peaked in the specular direction (Fig. 4.7) making the separation of the specular and diffuse components very difficult experimentally. This is a very difficult problem since the qz dependence of the diffuse intensity depends on the exact interface correlation function. A simple model can therefore no longer be used for the analysis of "reflectivity" curves. This is the situation found for the system of Fig. 4.8, an octadecyltrichlorosilane Langmuir film on water [32]. In this case, the surface spectrum can be calculated, and the specular and diffuse contributions to the reflectivity can be compared. The roughness spectrum (here thermally excited capillary waves) is obtained from thermodynamic considerations by Fourier decomposition of the free energy, see Chap. 9:
Equation (4.66)
above shows that the diffuse
.
(z(q11)z(-q11))
kBT
I -
L,
x
Ly zApg
2
4
+ -yq + Kq 11 11
(4.67)
L., x L. is the interfacial area, -y is the surface tension, and K is the bending rigidity modulus. The correlation function can be obtained by Fourier
Diffuse
4
Scattering
149
transformation.
(z(O)z(rjj)
--
kBT121r-y
[Ko(rjjVZ21p-g1 )
x
-
Ko(rjjV7--1K)],
(4.68)
where KO is the modified Bessel of second kind of order 0. Then, for a wave vector resolution Aq.,, the intensity measured in the 0,,, = Oi,, direction is smaller than the
7r- 1/2r
[2 I
reflectivity kBT q2
.
47r-y
z’
I
2
of
perfectly
a
Aq2
’
KI
x
flat interface
exp-
7
1
27r^/
In
by
(
a
factor:
e--1--
V/,-Y-/I,(
A/2-
Aqx
)I
4.69)
incomplete -V function, and -yE Euler’s constant. Note that _,2 ’ because diffuse scattering has been taken larger than e -q 2, in addition to specular reflectivity.
where F is the this factor is
,
into account
10-1
1
q7=1.5nrn-1 10-1
.11u=1t(n,) (0ff -1
z x
t=30nm
v=0.3
10-1 .S x
__P
-
-
.1
10-1
-
water
io-10
10-1 1
y=0.073 Nrn-1
11
0.03
L 1
L
-0.02
-0.01
0
0.01
0.02
0.03
% (nm
scattering from the water surface which is peaked in the specular capillary waves of longer wavelength cost less energy (only a calculated intensity is presented here because the large background due to bulk scattering prevents from a precise measurement, see below the chapter on liquid Fig. 4.7.
Diffuse
direction because
surfaces),
and
It
be
fuse
can
a
solid surface with
seen on
intensity
Fig.
a
4.8 that
flat power spectrum
even
for
relatively small wavevectors the difpossible to obtain physically
dominates. It would not have been
150
Jean Daillant and Anne Sentenac
101 10-1 10-2 10-1 10-4 ’I-10- 5
10-6 10-7 10-1 10-1 10-
V
10 r
0
2
4
q,
6
1
8
(nm-1)
Fig. 4.8. Reflectivity of an octadecyltrichlorosilane film on water. The broken line corresponds to specular intensity. It is dominated by diffuse intensity (grey line) for wavevectors larger than 2nm-’. The black line is the total (specular + diffuse) intensity. Inset: corresponding electron densities for the complete model Eq. (4.69) (thick line) and the simple box model with error function transition layers (thin
line)
reasonable parameters from the into account (see Fig. 4.8).
4.A
experiment without taking its contribution
Appendix: the Reciprocity Theorem
permittivity Cref (r) which is region of space. Let two different current distribution sources JA, JB (with same frequency w) be placed in this medium. We denote by the indices A, B the fields created by these sources, separately, in the medium. They satisfy Maxwell’s equations, We consider
a
medium described
assumed to be different from I in
V
X
V
X
EA(B) HA(B)
by
a
the relative
localised
=
=
-iwBA(B) JA(B) + iwDA(B)) (4.Al)
Diffuse
4
DA,(B)(r)
where
the Maxwell’s
--
COCref(r)EA(B)(r)
equations
’7.(EA
X
HB
-
and
X
HA)
HA(B)/Po.
--
151
Substituting
identity,
in the vectorial
EB
BA(B)
Scattering
HB.V
x
EA
EA.7
x
HB
HA-V
x
EB + EB.V
x
HA,
-
(4.A2) leads to V. (EA
X
HB
-
EB
X
HA)
:--
EA JB + iw (EB.DA
EB JA
+iw(HA.BB
The last two terms
on
V.(EA
X
Integrating Eq. (4.A4)
I and
d3,r,7. (EA the
using
now
X
HB
-
-
HA)
X
all space
over
HB
EB
-
FIB
(EB-JA
=
-
that
get,
we
EAJB)
EA.DB) HB.BA)(4.A3)
(4.A4)
-
gives
HA)
X
are zero so
-
I d3r (EB
-
JA
-
EA JB)
)
theorem
EB
x
HA)
-_
I
d3r (EB JA
-
EA JB)
-
(4.A5)
sources are limited to a finite volume, the surface of inteEq. (4.A5) is infinitely remote from them, and the electromagnetic be approximated by a plane wave with E and H orthogonal and
the current
gration field
HB
divergence
I d2 V(EA If
X
side
right-hand
the
-
in
can
transverse.
H
It follows
FW60-0-
fi
X
E.
that, EA
which
=
X
HB
-
EB
X
HA
-
0)
yields d 3rEA-JB,
d3rEB-JA
reciprocity theorem [12-14]. Eq. (4.A6) can (Iliw)J, one gets dipole density sources through P
which is the
(4.A6) also be written for
--
I d3rEB-PA jd3rEA-PB-
(4.A7)
152
Jean Daillant and Anne Sentenac
4.B
Appendix: Verification
of the
the Case of the Reflection
Integral Equation
by
Thin Film
a
on
in
a
Substrate It has been indicated in the niain text that the
integral equation Eq. (4.11) by applying the reciprocity theorem is an exact equation. In this appendix, we verify that this is indeed the case for a single film on a substrate. The reference situation an homogeneous medium of optical index no and we obtained
want to calulate the electric field in the
thickness d
the smooth film
r
are
2ik ,id roj + rl,26-
-_
ro,jrj,2e-2ik,,,1d
differs from the ideal
case
where there is
case
a
(1)
film
of
The reflection and transmission coefficients of
respectively, Eqs. (3.75), (3.76):
I +
The real
(2).
substrate
on a
10,14,2e_ ik
t
’
+
one
ro,jr1,2e-2ik ,1d
within the substrate where the
fractive index difference between the real and ideal 2
(n 1
the film where the difference is
_
n
2). 0
,jd
In
(n 22
is
case
_
n
2) 0
re-
and in
Eq. (4.11), we need the field in substrate, and is
the real case, which is the transmitted field in the PW
El
(Z)
to’l
=
1 + rojrl,26 2ikj_,jd
[e-illin,,11
in the film. We also need the Green function in
koe- ikoR 47r,EOR
Using Eq. (4-11),
the electric field
-
E
--
6
Eo +
ikoR
j drjj
47rR
can
-
10 koz ) f
e- k_’0Z
ko2z)
,
vacuum
I
(medium (0)),
ik_,0.r
be written:
jqjj.rjj
d
2
(klz
e
+ rl,2 e-ikj-,1(2d-z)
10, (e-ikin.,iz +
ikjn.,j(2d-z)
r12
+ roirl2e- 2i’kin.,Id
dz
d
2
+(k2z In
Eq. (4.Bl),
no2) to
=
kz2, j
ensure
-
we
kz2, 0,
e- kscz ’0zte- ikin.,2(z-d)
2
-
2,
have used that k z
Medium
(2)
The differential
-
Eo
e-ikR -
scattering
k?-k 2
=
is considered to be
the convergence of the
E
]1.
(4.131)
00
47rR
integration.
n k 02
k2
X
implies
k 02(n
slightly absorbing in
-
3
order
One obtains:
j drjj e’q11.rjj (2iko
cross-section is
-
thus:
sin
Oo r).
(4.132)
Diffuse
4
b2
do-
-0
And
we
Ir 12 Sin2
47r2
dQ
k 0 sin
2
00 1r 12Aj(qll).
153
(4.133)
find for the reflection coefficient:
R
as
2
Oo472Ajk- jj,kj ,,,
Scattering
LdS2
A sin 00
12
M
(4.134)
expected.
Roughness in Approximation
Interface
Appendix:
4.C
Within the Born
Multilayer
a
appendix we treat the case of a rough multilayer within the Born approximation in order to show some simple properties of the scattered intensity. In the case of the rough multilayer depicted in Fig. 4.2 Eq. (4.26) gives: In this
do-
ko Cos2
dz
drjj
167r 2
dQ
2
(n i
-
1)e iq.r
2
(4.Cl)
The upper medium (air or vacuum) is medium 0, and the substrate s) is slightly absorbing in order to make the integrals converge.
Let
Cos
167r2
dQ
which
8
k4 -0
do-
can
be written
dQ
167r 2 qz2
iq,,zi+i.
iqzj
_
iqjj.rjj
n?
(4.C2)
2
Cos
2o
1:
drjj In i+ 1 2
-
n?] 2
e
iq,,zieiqjj.rjj
(4.C3)
i=O
then define:
zi
where Zi is the can
height
=
Zi +
zi
(4.C4)
(r1j),
of the flat interface in the reference
case.
Equation
be written:
k4. 0
do-
dQ
e
iq,
N
ko
(4.C3)
1
as:
4
do-
us
1 drjj
2o
(medium
Mr 2 qz2 e
iq ,
N
COS2 0
X
N
E I: J drjj I dr’ll In i+ 2
i=O
1
-
n?] [n +, S
3
-
nfl 3
j=O
(Zj-Zj),iq , (zi(ril)-zj(r’ll)),iqll.(rll-rlll).
(4.C5)
Jean Daillant and Anne Sentenac
154
Making
of variables
change
the
0
-COS’
167r 2 qz2
dQ
zi(r1j),
(second) order,
we
over
r’ll:
N
2
-
S
,
nfl
-
3
3
j=O
,iq,, (zi(r1j)-zj (0)) ) z’qjj.rjj’
where A is the illuminated
fluctuations
r1l and integrating
-+
Y E j drjj [ni+l n?] [n +j
X
i=O
iq (Zi- Z5)
e
r’ll
-
N
k 4A 0
do-
r1l
area.
in any have: or
(4.C6)
Assuming Gaussian statictics of expanding the exponential to
case
(,iq,,(zj(rjj)-zj(O)))
-- 1q,2,:(zj(rjj )_Zj(o))2
-
the
height
the lowest
(4.C7)
We then obtain:
do-.
1,4 A Cos
-
dQ
161r2q., N
v)
N
11" E I i=O
n
nj2 [nj2+1 3
2
i+1
-
e’q- (Zi Zj) e 1q.2,(zj)"--Lq,2,(zj)’
nj2 ]
-
-
-
2
2
3
j=O
,q,2,(zj(rjj)zj(O))2,iqjj.rjj.
d rjj This
2
(4. C 8)
.
specular components because it has been congeneral solution of an electromagnetic field in a vacuum. intensity can be obtained by removing the specular component:
equation
also includes
structed from the
The diffuse
ko’A
do-
N
N
T Y 1: [n i+
dQ
2
n?] [n +,
i=O
-
3
,
aq _,
nfl 3
j=O
ei,j,(Zj-Zj)e- !q.2(,j)2- 1q.2 (,j)2 J(qll). 2
The diffuse
(dQ
intensity
is
then:
k4A 0
do-
__
incoh
16ir2q2 z
N
N 2
[ni+l i=O
Cos
For
has
(4.C9)
2
-
nj2] [nj2+1
-
nj2]
eiq (Zi-Zj)e- -g1q,2,(zj)2- 7q2,(zj)2
j=O
2o
j drjj (,q.2(zj(rjj)zj(O))
2 -
I
) eiqll.r1j.
(4.CIO)
single surface, we get Eq. (4.35). It is remarkable that equation (4.CIO) exactly the same structure as the reflectivity coefficient (Fig. 4.3),
a
4
do-
k0A
dQ
4qz
h
e
N-1 N-1
E 1: [ni+l 2
i=O
-
nj2] [nj2+1
-
nj2]
j=O
iq (Zi-Zj) .-
- 3q,2(zj)2- -g3q,2. (Zj)2 2
2
(4. C 11)
4
each term
simply being multiplied by I
47r 2
I
dr
a
Scattering
155
"transverse" coefficient:
(,q.(zi(rjj)zj(0))’
11
Diffuse
eiqlIx1j.
Appendix: Quantum MechanicalApproach Of Born and Distorted-Wave Born Approximations
4.D
T. Baumbach and P. Mikulfk
appendix we treat the formal quantum-mechanical approach to scattering by multilayers with random fluctuations. That can be interface roughthe ness, but also porosity or density fluctuations. In particular we develop kinematical in the section differential scattering cross approximation (first Born approximation) and in the distorted wave Born approximation in terms of the structure amplitudes of the individual layers and of their disturbances. This approach is written in a general way. In Chap. 8 it will be applied to the reflection and to diffraction under conditions of specular reflection under grazing incidence by rough multilayers and multilayered gratings. We would like to notice that we adopted here the phase-sign notation of this book, -ikr and Fourier transforms e +iqr, which is contrary to with plane waves that used in most publications using this formalism. In this
Formal
4.D.1
Theory
develop formally the incoherent approach for the scattering by multilayers with defects independently of the specific scattering method. We make use of the (scalar) quantum mechanical scattering theory and its approximations, in particular the first order Born approximation (kinematical theory) and the distorted wave Born approximation (semi-dynamical theory). Scattering of the incident wave JKO) by the potential V produces the total wave field JE), described by the integral equation [33]
Here
we
I E) where
00
transition
can
I Ko)
+
OJJE)
(4-1)1)
,
operator of the free particle. We define the I JE) and the transition rnatrix by the matrix
is the Green function
operator by
elements Tos The
--
=
tjKo)
=-
(Ks Itl Ko), characterising the scattering from I Ko)
differential scattering expressed by the matrix
cross
be
do-
section
a
into
an
into
I Ks).
elementary solid angle 6Q
elements of the transition matrix 1
=_
167r 2
JTos 12
dQ
.
(4.D2)
Jean Daillant and Anne Sentenac
156
Scattering by spatial
randomly
disturbed potential. including a random scattering potential, the differential cross section the statistical ensemble of all microscopic configurations a
fluctuation of the
averages
over
do-
(ITOS12
--
dQ
f
167r2
(4.D3)
We divide do- into coherent and incoherent contributions do-
1
(TOS) 12+
167r2
167r2
I COV(TOS’ TOS)12
dS2
=-
dO’coh
+
dUincoh
(4. D 4) by denoting the covariance
Cov(a, b) Defining the
potential)
=
(ab*)
(a) (b)*
-
(4.D5)
.
non-random part of the scattering potential by VA (unperturbed the random (perturbed) potential by VB’ the coherent part
and
of the differential
cross
section writes
dUcoh
1 -
167r 2
and the incoherent differential
dO’incoh
cross
A
IT
+
(T B) 12
(4.D6)
dQ
section
:::::::12 Cov(TB 167r
,
T
B)dQ
(4.D7)
If the random part VB causes only a small disturbance to the scattering B by VA, we can calculate T within the distorted wave Born approximation
(DWBA). is not
a
It is worth
noting that in contrast to the widely spread opinion it potential VB < VA which defines the validity of the DWBA, the scattering by VB which has to be weak.
small
but rather
Scattering by
randomly
a
disturbed
multilayer.
In
a
multilayer
represent each layer by the product of its volume polarizability the
X,,,, j
(r)
we
and
layer size function Qj(,r) N
X
1: Xj (r) j=1
N -
1: X. j H Qj (r)
(4.D8)
j=1
optical (or scattering) potential for X-rays can be expressed by the polarizability: V(r) -ko2X(,r). The contribution of the different layers to the section is distinguished by considering each layer as an indescattering cross scatterer pendent The
=
Vj (r)
(4.D9)
Diffuse
4
N
I
do-
E
--
j)
12
with Tj
(Ks I vj I E).
=
Separating
do-
vjB,
jE
3
j
(,rj, Tk)
Cov
(4.DIO)
dQ
,
N
N
TA j
+
1: (’rB) 12 j
+
j=1
j=1
where the r-:1 are
j=1
k=1
part of each layer, vj
the non-random and the random
N
i-6jr2
and TB
N
obtain
we
I =
N
+
j=1
vjA
157
Eq. (4.D4) writes
Then
+
Scattering
EE j=1
the contributions of the
are
those of the
N
Cov
(,rB,,FB k ) j 1;
(4.D11)
dQ
k=1
non-perturbed layers
to
scattering,
term is the coherent
layer disturbances. The first
part dUcoh; which consists of the contribution of the ideal multilayer and of the
averaged
transition elements of the
layer disturbances. The second, single layer
incoherent part do-incoh contains the covariance functions of all transition elements.
Formally the division of ’ into a sum of scatterers E, vj is arbitrary. The sticking point is to find a set of eigenstates, which is convenient to serve as basis for calculation, of the transition elements. Finally, we remind the reader, that until now no approximation has been made.
4.D.2
Formal Kinematical Treatment
by First Order
Born
Approximation
Within the kinematical treatment
(first
approximation)
order Born
we
ap-
scattering potential flK) P_- ’JK). The set of vacuum wave vectors JK) e-ikr provides an orthogonal basis for the calculation of the differential scattering cross section. The transition elements of the individual layers are proximate
by
the transition operator
the operator of the -
Tj
where q
--
ks
-
=
(KsjvjjKo)
ko. Defining the
-k02
I
structure
Sj (q) with the random one-dimensional
Fj(qz, r1j)
=
dr Xj
(r)
factor of
drIl Fj (q, r1j)
e
iqr
the
ez q1I r1l
(4. D 12)
layer
(4.DI3)
layer form factor dz Xj
(r)
iq, (z
-
Zj)
(4. D 14)
Jean Daillant and Anne Sentenac
158
the transition element becomes
-k20 eiq.,Zj S (q) The coherent
(7j)av,
and
scattering
cross
search for the
so we
incoherent differential
scattering
section
(4.D15)
(4.D9)
uses
the statistical averages The
layer form factor
mean
cross
Fj (q,,,rjj))av.
section contains the covariance
func-
tions
COV(SjSk) P
I
d’rjj
I drii’ciqll(rll-rll’)COV(Fj(q,,,rll),Fk(q,,,rll’))
Substituting (4.D13) scattering
differential
4.D.3
(4.D16)
and cross
1 1: (Sj) ’V
MT 2
iq ’ Zj
e
3.=I
12
(4.1)10),
into
section of
N
00
do-
(4.D16)
an
we
obtain the kinematical
arbitrary multilayer
N
N
E E & e’q;, (zi
+
j=1
-
ZO
dQ
.
(4. D 17)
k=1
Formal Treatment
by
a
The distorted
Distorted Wave Born Born
wave
Approximation
approximation takes all those effects of multiple are caused by the unperturbed potential V’.
into account which
scattering
right choice of VA, which decides enough transparent and sufficiently precise. We search for such a VA which enables to explain the essential multiple scattering effects. However, it should provide the simplest possible solutions EA used as orthonormal basis for the representation of scattering by the K disturbance (perturbed potential) VB. It is less the method
about the
itself, but
rather the
in order to be
success
Scattering by planar multilayers with sharp interfaces produces such simple solutions. It has been shown that rough multilayers as well as intentionally laterally patterned multilayers and gratings can be treated advantageously by starting with an ideal potential of a planar (laterally averaged) multilayer, splitting the polarizability in N
X
=
X
A
+ X
B
with
X
A
E XiAplanar
(4. D 18)
j=1
Coherent
scattering by the non-perturbed multilayer generates a wave field can be decomposed into a small number of plane waves within each plane homogeneous layer, both with constant complex amplitudes and A EK which ,
wave
vectors, A
EK ’i (r)
Ek,,,j
-
n=1
e- ik"jj’jTjj e- ik,_,,j(z-Zj)
S2A (Z) j
(4. D 19)
Diffuse
4
In
of
case
specular reflection
wave), for grazing tion 1=8. The
is,
it
transmitted and
(one
1=2
Scattering one
159
reflected
strongly asymmetric X-ray diffracnon-perturbed states for the estimation of
incidence diffraction and
A
EK (r)
are
used
as
TB (4.D6). Within the first order DWBA
TB,DWBA Again,
=
(E SA*IfrB JE 0A)
obtains
one
-k
-
-
0j
A
2
dr E S
A
(r )XB (r)E 0 (,r)
.
(4.D20)
it is recommendable to describe the contribution of the disturbance
within each
plane layer separately by B
73
-
2
A B E’ S (r) X3 EO’ (r)
dr
-k 0
(4.D21)
with Bplanar
j
(4.D14),
however
to the actual
now
scattering
expressions (4.D13) and corresponding
vector
km
=
Sj
-
(4.D23)
kn
Oj
layer
Fjm’(q’ 13’, r1j)
dz
=
Z
consists of I
Each
(4.D22)
Xj
with respect to the disturbance X)P and
q,mn inside the
-
similar to the
Sj"’, formally
We define Fmn and
Aplanar
XQA j
Xj
x
-
-
Zj)
(4.D24)
I terms
2
B
Tj
XP (r) e’q--, in (z 3
Em
-k 0
Sj
(4.D25)
n
E (Z) (Z)Smn Oj j
m=1 n=1
or
using
the matrix formalism B
Tj
-koEsj jEoj 2
-
rn
contains the where the column vector En Kj
(4.D26)
n
amplitudes of the
I
plane
waves
of
non-perturbed state in the jth layer and j is the structure factor matrix layer disturbance, respectively. Each term in (4.D25) represents the contribution of the disturbance to the scattering from one plane wave of the A A in another plane wave of the final state EK,. Each scattering initial state EKo of the according wave amplitudes process is characterised by the product
one
of the
E’ En Sj
03
and by the disturbance structure factors Smn. j average F the statistical ensemble and substitute these terms in (4.DIO).
In order to determine the coherent scattering
and S
over
cross
section
we
Jean Daillant and Anne Sentenac
160
The incoherent
section contains the covariance functions for each
cross
layer
pair Cov
B
B’
(7-j
4
E, m _E-n
ko
7k
Sj
S,
’P Qmn jk
E E’ Ok
(4.D27)
Ok
m,n,o,p
with OP Qmn jk
Cov
QOP (STn 5-k J 3
j drjj j drll’eiqjj(rjj-rjj’)
COV
(Fj (qm ’, r1j), Fk (ql
z,k
Z,3
,
(4.D28)
r1l
Each term represents the covariance of one scattering process in layer j and second scattering process in layer k. Adding up the contributions of all
a
scattering
processes and all
layers
obtain
we
finally 2
k 0’
do-
E Tj
167r2
+
1: E
EM
Sj
En (Smn) j Oj
(4.D29)
j=1 m,n=l
=1
N
I
E"
+
Sj
(E-Sj )* Qmn,, ik
* E, Ok (Epok)
dQ
j,k=l m,n,o,p=1
X-ray reflectivity, each eigenstate of the unperturbed potential consists
In
of a transmitted and reflected wave, thus _T
qll,. (8.48),
q22, corresponding
..,
to
(kscll
-
=
2. The four
kin1j; k,,c, ,
wave
ki,,,,)
in
vector transfers
(4.46),(4.47)
or
represented in the reciprocal space in Fig. 8.40. Further, The above expressions are written explicitly for diffuse scattering in Eqs. (8.46)(8.49) and for coherent reflectivity for deterministic (i.e. non-random) grating potential VB in (8.72). The covariance for grazing incidence diffraction is are
presented by (8.62).
Stmpler
DWBA
for multilayers.
The
expressions simplify enormously, approximate non-perturbed polarizability by its mean value in the multilayer, averaging vertically over the whole multilayer stack. We obtain a homogeneous "non-perturbed layer". The splitting of the potential in this way gives if
we
the
can
XA (1,)
-
(XML (r))
av
N
X, (r)
-
E XiBlaper (r)
with
XiBlay-(,r)
=
(X(,P)
_
(XML(,r)) ) oi.d(,r) av
j=1
(4.D30) Now the
non-perturbed wave only. In
transmitted
wave
field below the
consequence
sample exclusively the primary scattering prosurface consists of the
cesses
’Cov ( Tj!3 ;
TB) k
K4 tSt*S
’ ror*0
3k
(4.D31)
Diffuse
4
and the transmission function of the
sample
surface
are
Scattering
161
considered. Also the
effect of refraction is included.
References Sinha, E.B. Sirota,
S.K.
1.
Stanley, Phys.
S. Garoff and H.B.
Rev. B 38, 2297
(1988). B61orgey, J. Chem. Phys. 97, 5824 (1992). Karabekov, I.V. Kozhevnikov, B.M. Alaudinov, and V.E. Atyukov, Asadchikov, Physica B 198, 9 (1994). 4. S. Dietrich and A. Haase, Physics Reports 260, 1 (1995). 5. D.K.G. de Boer Phys. Rev. B 49 5817 (1994). 6. P. Croce, L. N6vot and B. Pardo, C.R. Acad. Sc. Paris 274 B, 803 (1972). 7. P. Croce, L. N6vot and B. Pardo, C.R. Acad. Sc. Paris 274 B, 855 (1972). 8. P. Croce and L. N6vot Revue Phys. Appl, 11, 113 (1976). 9. P. Croce J. Optics (Paris) 8, 127 (1977). 10. L. N6vot and P. Croce Revue Phys. Appl. 15, 761 (1980). 11. P. Croce J. Optics (Paris) 14, 213 (1983). 12. P. Lorrain and D.R. Corson "Electromagnetic Fields and Waves" W.H. Freeeman and Company (San Francisco) (1970) p.629. 13. L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media, Course of theoretical physics vol. 8, Pergamon Press, Oxford 1960, 69. 14. C.-T. Tai, Dyadic Green functions in electromagnetic theory, IEEE Press, New2.
J. Daillant and 0.
3.
I.A.
A. Yu.
’
York,
1994.
15.
J.D. Jackson "Classical Electro dynamics" 2
16.
M. Born and E.
Wolf, "Principles
of
d
optics"
Wiley (New-York) 1975. edition, Pergamon (London)
Edition 6
th
(1980) p.’51. Sinha, M. Tolan. A. Gibaud, Phys. Rev. B 57, 2740 (1998). Herpin, C. R. Acad. Sci. Paris 225 182 (1947). Y. Yoneda, Phys. Rev. 131, 2010 (1963). B. Vidal et P.. Vincent, Applied Optics 23 1794 (1984). I.M. Tidswell, T.A. Rabedeau, P.S. Pershan, S.D. Kosowsky, Phys. Rev. Lett.
17. S.K. 18. 19. 20. 21.
A.
66, 2108
(1991).
Waifan, J. Appl. Phys. 3 503 (1970) Cai, K. Huang, P.A. Montano, T.P. Russel,.J.M. Bai, and G.W. Zajac, J. Chem. Phys. 98 2376 (1993). 24. W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992). 25. A. Sentenac and J.J. Greffet, J. Opt. Soc. Am. A 15, 528 (1998). 26. G.C. Brown, V. Celli, M. Coopersmith and M. Haller Surface Science 129 507
22.
J.B. Bindell and N.
23.
Z.-h.
(1983) Brown, V. Celli, M. Haller and A. Marvin Surf. Sci., 136 381 (1984). Brown, V. Celli, M. Haller, A.A. Maradudin, and A. Marvin, Phys. Rev. B,
27. G.C. 28.
G.
31 4993
(1985).
in volume and surfaces, ElsePublishers, B.V. North-Holland (1990) 30. R. Petit, ed, Electromagnetic theory of gratings, Topics in current physics, Springer Verlag, Berlin (1980). 31. W.H. de Jeu, J.D. Schindler, E.A.L. Mol, J. Appl. Cryst. 29 511 (1996). 29. M.
Nieto-Vesperinas and J.C. Dainty, Scattering
vier Science
Jean Daillant and Anne Sentenac
162
32. L.
Bourdieu, J. Daillant, D. Chatenay, A. Braslau, and D. Colson, Phys. Rev. 72, 1502 (1994). A.S. Davydov, Quantum Mechanics, Pergamon Press, 1969.
Lett. 33.
1 149
(1991).
Neutron
5
Claude 1
Ferman’,
Reflectometry
Fr6d6ric Ott’ and Alain Menelle’
Physique de I’Etat Condens6, Orme des Merisiers, CEA Saclay, Yvette Cedex, France, 2Laboratoire Uon Brillouin CEA CNRS, CEA Saclay, 91191 Gif sur Yvette Service de
91191 Gif
sur
Cedex, France
Introduction
5.1
technique [1,2]. In the last years, problems like polymer solving extensively surface the of at the structure [5,6] for example. The liquids mixing [3,4] or small their is studies for absorption compared asset of neutrons polymer
Neutron
reflectometry is
a
relatively
new
used for
it has been
to x-rays and the
"labelling" by
large
contrast between
soft matter
’H and 2H which allows selective
deuteration.
80’s, a new field of application of neutron reflectometry has emerged. Following the discovery of giant magnet oresist ance in antiferromagnetically coupled multilayered -films [7] and new magnetic phenomena in ultra-thin films, there has been an interest in the precise measurement of the magnetic moment direction in each layer of a multilayer and at the interface between layers. Owing to the large magnetic coupling between the neutron and the magnetic moment, neutron reflectometry has proved to be a powerful tool for obtaining information about these magnetic configurations and for measuring magnetic depth profiles. In this chapter, we give an overview of the experimental and theoretical methods used for neutron reflectometry, focusing on specular reflectivity. The corresponding theory is partly derived from the previous work developed for of neutrons. x-rays, and we emphasize those aspects specific In the late
In
a
an
will review the neutron-matter interactions. We then non-magnetic scattering. In this case it is possible to introduce
first part,
describe the
optical
we
index and
give
a
treatment which is similar to x-ray
reflectometry
(Chap. 3). In a second part, the neutron spin is introduced. In this case, optical indices cannot be used any longer and it is necessary to completely solve the Schr6dinger equation. A detai,led matrix formalism is presented.
We then discuss the different aspects of data processing and the problems roughness. Two types of neutron reflectometers are
related to the surface
particular: fixed- wavelength two-axis reflectometers and timeof-flight spectrometers. The use of neutron reflectivity in the field of polymers films and of magnetic layers is then illustrated by several examples.
described in
J. Daillant and A. Gibaud: LNPm 58, pp. 163 - 195, 1999 © Springer-Verlag Berlin Heidelberg 1999
Claude Fermon et al.
164
Notation used in this
b, bj
bound
chapter
scattering length of
a
nucleus,
mean
scattering length
layer j b c*
bi bN V V
90, Sj 6
bound coherent
scattering length scattering length spin dependent scattering length real part of the scattering length imaginary part of the scattering length incoherent
energy of the neutron in the charge of the electron
d, dj
thickness of
9
Land6
Planck constant nuclear
k
wave
spin operator
vector
M, Mj magnetic
moment of
neutron
mass
M,
electron
mass
nj
refractive index of
an
electron and of
layer j conversion factor 2.696fm, p an effective scattering length scattering vector spin operator of the electron --
q S.
of
a
Pauli operator associated to the neutron
Vi
volume of the
V(T)
interaction Hamiltonian
gn
gn
A, A0
neutron
PB
Bohr magneton nuclear magneton
Pn
layer
magnetisation
0-
--
layer j
layer factor, (g =2)
I
P
and in
a
h
M
vacuum
to
spin
layer j
-1.9132, nuclear Land6 factor wavelength
of the neutron
density of the layer j (atoms per cm 3) absorption 0’j Oj, oj spherical angles of the magnetisation of the layer j Oin, Or incident and reflected angles of the neutron beam atomic
Pi
PB
Pn
=
=
O(x)
eh/(2m,) eh/(2mp)
is the
=
=
9.27
x
5.05
x
10-24 J.T-1 10-27 J.T-1
Heavyside function defined by:
O(X) O(x) O(X)
=
I
=
1/2
=
0
when
x
when
when
> x
x
0
0.01
0.01-
Thdaiess (rui
0.00 0
Mida-ess (m 0.001-
0.001-
o
0.0001
V
0.00M
o
-
o
oo
0.00001 0
q
q
(14
0.3
0.2
0.1
0
0.4
O’
(nnrl)
oo
o."I
0.00001
0.2
0.1
oo
(Poo 1
1
(nnrl)
Left: Reflectivity of a quadrilayer consisting in a partially deuterated copolymer layer deposited on a trilayer of totally hydrogenated polymer (measured on the EROS reflectometer at the Laboratoire L6on Brillouin.) Right: Reflectivity of the quadrilayer after annealing for I hour at 115’C
Fig. 5.11.
PS-PBMA
5.8
Examples
on
Magnetic Systems
give some examples in order to highlight the information that can be obtained by polarised neutron reflectometry. All the experiments shown here have been performed on the reflectometer PADA.
In this part,
shall
Absolute Measurement of
5.8.1 Neutron
atom).
we
reflectometry
can
be used to
sample. As an NiFe single layer. By fitting Fe with
an error
Magnetic
measure
independent example, Fig.
The obtained value is
surface of the
about
a
Moment
absolute moments
of the
(in
1-tB per
layer thickness and of the
5.12 shows
a curve
obtained
on a
the curves, we obtain the ratio between Ni and about of 2% and the absolute moment with a precision of
0.02/-tB. That
measurement took
only
15 minutes
on a
I
cm
2
sample.
Claude Fermon et al.
190
1000000
100000
10000
1000
100 0
0.2
0.4
0.6
q
Fig.
reflectivity R++ intensity and white 5.12.
5.8.2
curve
Bragg
on
a
25
squares R--
Peaks of
nm
0.8
1
1.2
1.4
(nm )
thick NiFe
layer. Black
squares
are
the
intensity
Multilayers
Periodic
Multilayer In the case of periodic multilayers, we can observe Bragg peaks corresponding to the period of the multilayer. In the case of antiferromagnetic coupling or variable angle coupling, it is possible to obtain directly a mean angle between the different magnetic layers. With polarised neutrons, it is possible to measure very rapidly a precise value of the average moments. If high"order Bragg peaks are observed, a good estimate of the chemical and magnetic interface can be obtained. In the literature, there is a large amount of results on magnetic multilayers [18,23,24]. 1000000
100000
10000
1000
100
10 0
0.02
0.04
0.06
0.08
q
Fig. spin
Example of a polarizing mirror. The two curves correspond respectively "up" (black squares) and "down" (white squares) state of incident neutron
5.13.
to the
Reflectometry
Neutron
5
191
Supermirrors [25] For technical purposes it is interesting to build systems exhibiting an articially large optical index. One can can build such a structure by stacking periodic multilayers with an almost continuous variation of the period. In such a system, if the periodicity range is well choosen, a large number of Bragg peaks follow the total reflectivity plateau. Since the periodicity of the multilayer is varying continuously, all these Bragg peaks add constructively. Using this technique it is possible to enhance the length of the total reflection plateau by a factor 3 to 4. Such mirrors are now widely used for neutron guides and for polarisation devices.’Figure 5.13 gives an example of a polarising mirror. Measurement of the In-Plane and Out-Of-Plane Rotation of Moments. Measurement of the Moment Variation in a
5.8.3
Single Layer perhaps the most important information given by polarised neutron reflectivity (PNR) for magnetic thin films. We shall give here two examples of determination of in-depth magnetic profiles.
This kind of measurement is
1000000
100000--
8 ML
Alloy (air/Au) d-down
Au
10000--
6.3
(111)
nm
5 ML
Alloy (Co/Au)
15
1000P-rip
101
-
100
-
__
-
-
-
-
-
-
-
-
R
-
0o
i
0
".
1.5
1
0.5
0
1000000
hcp (0001).
Co
%
2
Alloy (Co/Au)
flat
(degreE!s) Au
100000
(111)
28.4
Alloy (air/ float glass) Float
100001
nm
5 ML
glass
1000
100
10 i 0
1.5
1
0.5 0
2
(degrees)
Reflectivity curve of a Au/Co(3nm)/Au trilayer system. Empty squares: reflectivity; filled squares: down-down reftectivity; lines: best fits. Top left: fit assuming a uniformily aligned magnetisation through the layer. Bottom left: fit with a model allowing magnetisation rotation; this model provides a better fit than the model using a uniform magnetisation. Right: Thicknesses and moment
Fig.
5.14.
up-up
directions
giving
the best fit
parameters
Claude Fermon et al.
192
Out-Of-Plane Rotation of the Moments in
Au/Co/Au [26].
In the
of very thin cobalt layers, thinner than 2.5 mn, the magnetisation is perpendicular to the surface layer. We have studied a 3 nm thick Co layer case
sandwiched between two Au
layers. Magneto-optic Kerr effect measurements mainly in-plane but with a small out of order to understand the magnetic behaviour of such a
have shown that the moment is
plane contribution. In layer, we have fitted the one,
we
PNR
curves
with two different models: for the first
have considered that the whole
second one,
reflectivity
have allowed
we
curves
and the
a
layer
rotation in the
corresponding
uniformly aligned. In layer. The Fig. 5.14 gives
was
the the
model.
Rotation in Strained microstructure
throughout
Nickel Layers In single magnetic thin films, the magnetoelastic (ME) properties vary sample: a gradient of the ME coefficient B(z) can appear, be such that the
can
the
related to surface relaxation effects. It surface
anisotropy
constants
B(z) where
is the
can
be written in
a
form similar to
[27]: =
Bbulk +
(z
-
(5.87)
ZO)
depth
in the thin film and zo an adjustable parameter. When a applied on a magnetic thin film, the magnetisation tends to rotate either along or perpendicular to the applied strain. A ME coefficient gradient will then lead to a gradient of magnetisation rotation through the thin film. This has been measured on single nickel layers as illustrated on Fig. 5.15. The numerical fit shows that the average rotation under a 0.03% deformation is 30’ but there is a 15’ gradient of rotation between the surfaces z
mechanical strain is
and the bulk of the material
elongation strain
03"a%,
1000000
[28].
!-..wn--D.wn -
’%k
116
field
’P_Up
Up-Up
M
’I -U Up P -Down
100000
Down-Down o 75*
glass substrate
Oo
-o
10000
-----------
100’
T
-Z
Nickel
(40 run)
100 C)
0.2
0.4
0.6
0
Fig..5.15.
Left:
0.8
1
1.2
1.4
(&gr-)
Reflectivity curves on a strained nickel layer (thickness 40 nm) for spin. The deformation applied to the substrate is 0.03 %. Right: Diagram showing the magnetisation rotation gradient in the strained 40 nm Ni layer deduced from the neutron fit each state of neutron
Neutron Reflectometry
5
A
Hysteresis Loops
Selective
5.8.4
complete
hours to be
193
reflectivity performed. If we
set of
layers as a function too long compared
of field
or
curves
(R++, R--
and
R+-)
takes about 12
magnetisation of different total experiment would be far
want to follow the
temperature, the
to the time
usually allocated
on a
neutron reflectometer
week per year). So the idea is the following [29]: from the fits of (typically the reflectivity curves in the saturated state we know the different thicknesses one
and
magnetic
multilayer system. We
states of the
are
then able to calculate
magnetic perform complete reflectivity curves for each value of the magnetic field, but we can measure only the reflectivity for (n + 1) well chosen 0 values where n is the number of different magnetic layers. Comparison of the experimental values obtained and calculations using reflectivity curves layers. It is then not the
for different module and orientation of the
necessary to
the parameters obtained from the saturated state allows us to rebuild the magnetic evolution as a function of the applied external field.
(gB/atome)
moment 2
T-
-6
3
0
-3
field(mT)
Fig. uous
5.16.
Hysteresis loops measured and from the reflectivity
curve)
squares,
0.4’ black
for
a
single
cobalt
measurements at
layer : by MOKE (continonly one angle (0.3’ white
squares)
1.6 1.4
.2
1.2
0.8 0.6 0.4 0.2 0 10
5
H
Fig. 5.17. Magnetisati6n
/Co(3.8A) /Pt(32A)
of each
(kG)
magnetic layer
of
thin film. The white squares
a
/Pt(98A) /Co(7.6A) /Pt(33A)
correspondto
the thinnest
layer
Claude Fermon et al.
194
Figure 5.16 gives the example of an hysteresis loop obtained for a single magnetic layer. Figure 5.17 corresponds to a more complicated system: substrate
/Pt(98A) /Co(7.6A) /Pt(33A) /Co(3.8A) /Pt(32A).
Such a technique layer separately. The sum of the two magnetisations agrees well with the magnetisation given by conventional measurements and the saturation value of each layer corresponds to the values measured on other samples with just one layer. has
given
5.9
the moment of each
Conclusion
on
Neutron
Reflectometry
This
chapter has given an overview of the neutron reflectometry as a tool for investigation of surfaces. We have presented a matrix formalism which makes it possible to describe the specular reflectivity on non-magnetic and magnetic systems. Neutron reflectivity is especially suited for polymer and magnetic thin film systems. This has been illustrated with a few typical examples. We have not given here examples of non-specular and surface diffraction experiments. This kind of experiment has suffered until now from the lack of intensity on the neutron spectrometers. Moreover, the formalism necessary to analyse the experiments in the case of magnetic surface diffraction is still being developed. The neutron has a good energy for inelastic scattering on condensed matter but we have not spoken here on this aspect of reflectometry which is rather new. A beautiful example of inelastic scattering is the measurement of the Zeeman energy [30,31]. The problem of phase determination in neutron reflectometry is also an active field of research [32-34]. If not only the intensity but also the phase of the reflectivity could be measured a direct inversion of the reflectivity profile would be possible. the
References 1.
2.
3. 4.
G.P. Felcher, R.O. Hilleke, R.K. Crawford, J. Haumann, R. Kleb and G. Ostrowski, Rev. Sci. Instr, 58, 609 (1987). C. F. Majkrzak, J. W. Cable, J. Kwo, M. Hong, D. B. McWhan, Y. Yafet and J. Waszcak, Phys. Rev. Lett. 56, 2700 (1986). J. Penfold, R.K. Thomas, J. Phys. Condens. Matter 2, 1369-1412 (1990). T.P. Russel, Mat. Sci. Rep. 5, 171-271 (iggo); T.P. Russel, Physica B 221, 267-283
5.
6.
7.
8. 9. 10.
11.
(1996).
Lee, D. Langevin, B. Farnoux, Phys. Rev. Lett. 67, 2678-81 (1991). J. Penfold, E.M. Lee, R.K. Thomas, Molecular Physics 68, 33-47 (1989). M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friedrich, J. Chazelas, Phys. Rev. Lett. 61, 2472 ( 1988). V.F. Sears, Physics Report 141, 281 (1986). X.L. Zhou, S.H. Chen, Physics Reports 257, 223-348 (1995). H. Glittli and M. Goldman, Methods of experimental Physics, Vol 23C, Neutron Scatterinj (Academic Press, Orlando, 1987). S. Dietrich, A. Haase, Physics Reports 260, 1-138 (1995). L.T.
Neutron
5
Refiectometry
195
Sears, Methods of experimental Physics, Vol 23A, Neutron Scattering (Academic Press, Orlando, 1987); V.F. Sears, Neutron News 3, 26 (1992). J. Lekner, Theory of reflection of electromagnetic and particle waves (Martinus Nijhoff, Dordrecht, 1987). S.J. Blundell and J.A.C. Bland, Phys. Rev. B 46, 3391 (1992). C. Fermon, C. Miramond, F. Ott, G. Saux, J. of Neutron Research 4, 251
12. V.F.
13.
14. 15.
(1996). 16.
Z., Physica B 94, 233-243 (1994). Croce, Revue de Physique Appliqu6e 15, 761 (1980). C.F. Majkrzak, Physica B 221, 342-356 (1996). B. Farnoux, Neutron Scattering in the 90’1 Conf. Proc, IAEA in Jiilich, 14-18 january 1985, 205-209, Vienna, 1985, X.D. F. Mezei, Z. Phys. 255, 146 (1972). T.P. Russel, A. Menelle, W.A. Hamilton, G.S. Smith, S.K. Satija and C.F. Majkrzak, Macromolecules 24, 5721-5726 (1991). X. Zhao, W. Zhao, X. Zheng, M.H. Rafailovich, J. Sokolov, S.A. Schwarz, M.A.A. Pudensi, T.P. Russel., S.K.Kumar and L.J. Fetters, Phys. Rev. Lett. Pleshanov
17. L. N6vot and P. 18. 19.
20.
21.
22.
69, 776 23.
YY.
(1992).
Huang,
G.P. Felcher and S.S.P.
Parkin,
J.
Magn. Magn. Mater. 99,
31-38
(1991). Schreyer, J.F. Aukner, T. Zeidler, H. Zabel, C.F. Majkrzak, M. Schaefer Gruenberg, Euro. Phys. Lett., 595-600 (1995). P. B6ni, Physica B 234-236, 1038-1043 (1997). E. Train, C. Fermon, C. Chappert, A. Megy, P. Veillet and P. Beauvillain, J. Magn. Magn. Mater. 156, 86 (1996). 0. Song, C.A. Ballentine, R.C. O’Handley, Appl. Phys, Lett. 64, 2593 (1994). F. Ott, C. Fermon, Physica B 234-236, 522 (1997). C Fermon, S. Gray, G. Legoff, V. Mathet, S. Mathieu, F. Ott, M. Viret and P. Warin, Physica B 241-243, 1055 (1998). G.P. Felcher, S. Adenwalla, V.0. De Haan and A.A. Van Well, Nature 377,
24. A.
and P.
25. 26.
27. 28. 29.
30.
409 31.
G.P.
(1995). Felcher,
494-499 32. 33.
34.
S.
Adenwalla, V.0. de Haan, A.A.
van
Well, Physica
B 221,
(1996).
Majkrzak and N.F. Berk, Phys. Rev. B 52, 10827 (1995). C.F. Majkrzak and N.F. Berk, Phys. Rev. B 58, 15416 (1998). J. Kasper, H. Leeb and R. Lipperheide, Phys. Rev. Lett. 80, 2614-2617 (1998). C.F.
Statistical
6
Alberto
Physics
at
Crystal Surfaces
Pimpinelli
LASMEA, Universit6 Blaise Cedex, France
Pascal
-
Clermont 2, Les
C6zeaux,
63177 Aubi6re
1
Thermodynamics
Surface
6.1
Surface Free
6.1.1
Energy
According to thermodynamics, physical properties can be deduced from the knowledge of the free energy. In this lecture, the surface free energy is introduced in a simplified way (disregarding, in particular, elasticity). In order to create a surface, one has to break chemical bonds, and this costs energy. At finite
temperature, the free energy has
to be considered. It is
easy to give, a precise definition of the surface free energy: to break a crystal along a plane, it requires a work W. If L’ is the area of the crystal section,
the surface free energy for unit area or surface tension is ao W/(2L2), the factor of 2 coming from the two surfaces which are created this way. It is
straightforward
area
to
see
that the number of broken bonds per unit surface particular, in the broken-bond
varies with the surface orientation. In
approximation compact than open
surfaces
are
expected
to have
a
larger
surface tension
that, (Problem, temperature.) Prove
for
instance, crystal, Given this dependence on orientation, if the surface is not a plane the total surface free energy may be expected to be the integral of the energies of all surface elements. This is only true for an incompressible solid of large enough size (the interested reader will find a discussion of this statement in o-(111)
1.155o-(Ool)
o-(111)
[1].
ones:
>
for
o-(ool)
an
>
o-(11o)
fcc
at low
If x, y and z are Cartesian coordinates and z(x, y) is the height of the over the xy plane, the local surface tension is then a function o-(p, q)
surface of the
partial derivatives P
(9Z1,9X,
q
azlay
representing the local slopes of the surface profile. The total free large, incompressible solid body containing N atoms is then: F
--
Fo +
A.
on the book "Physics of crystal Villain, Cambridge University Press (1998). The should refer directly to it for delving deeper into the different
Pimpinelli
interested reader
a
I I o-(p, q)dS
Most of the material in this lecture is based
growth",
energy of
subjects quickly treated
& J.
here.
J. Daillant and A. Gibaud: LNPm 58, pp. 199 - 216, 1999 © Springer-Verlag Berlin Heidelberg 1999
Alberto
200
Pimpinelli
where the first term on the right-hand side is the bulk free energy, and the second term is the surface free energy. The integration is made over the surface of the solid and dS is the surface element. For
an
incompressible solid,
F(T, V)
one can
equivalently
use
the
(Helmholtz)
the Gibbs free energy (free enthalpy) G(T, P) PV. The latter is known from thermodynamics to be equal to yoN
energy
or
thermodynamic limit constant pressure we
shall
N
and in the absence of
-+ oo
adequately described
are more
surface).
a
--
free F +
(in
the
Processes at
if G is used. In the
future,
the free energy F because, in the absence of external it results directly from the interactions between the molecules of the
mainly
use
forces, solid. Sometimes,
we
will
drop the
where energy and free energy In formula
(6. 1),
are
word
"free", especially
at low
temperature
not very different.
the free energy is assumed to be independent of the only true for a large sample. (Problem. Why? Is the .
local curvature. This is
assumption appropriate
for
membranes?)
Another
simplification
comes
from
the fact that the surface itself may be difficult to define: its location may be rather imprecise. This point has been addressed by Gibbs [2]. Also, in formula
(6. 1) an
z (x, y) is analytic, which implies that fluctuations on ignored-or, better, averaged out (what is called coarse-
it is assumed that
atomic scale
graining
are
in the
jargon).
Free
6.1.2
Step
At
temperature,
zero
Energy
a
(001)
or
(111)
surface is flat. At
thermal fluqtuations will make atoms leave the surface
higher temperatures, plane and create holes
and mounds. In other words, steps are present on the surface at any finite temperature. In the same way as the surface free energy, a step free energy may be defined. This is how to proceed: a bulk solid is cut into two pieces of
cross
section L
x
L, but this time
a
step is also
cut. The work needed to
take apart the two pieces is proportional to the total surface exposed, which is now L2 + U, if b is the step heigth. The total free energy is thus W 0"-
b
-
L2
-
0" 0
(6.2)
+7L
The "excess" free energy due to the step, ^/, is called the step free energy or step line tension. If more steps are present, (6.2) shows that the total free energy is
proportional
to the number of
steps
per unit
length,
or
step density,
the steps are far enough from each other that their interactions can be neglected. Broken bond arguments show that the step free energy is a function of the step orientation. as
long
as
6
Singularities
6.1.3
Statistical
Physics
at
Crystal Surfaces
201
of the Surface Tension
Let us suppose that the plane z -- 0 is parallel to a high-symmetry orientation, for instance (001), and that the temperature is low. We will see the precise
temperature" when addressing the roughening transition experimental realisation of a flat high-symmetry a certain length. In other words, real surfaces are always slightly "misoriented" or "miscut" with respect to a given highsymmetry plane, so that they make a small angle 0 with that orientation (a typical lower bound for 0 is 0.1’). As said above, if step-step interactions are neglected-which is licit, if steps are rare-the projected surface free al cos 0 is simply equal to the step free energy Multiplied by the energy 0 step density I tan 0 1:
meaning
of "low
in Sect. 6.2. Of course, no surface is possible beyond
-
--
0(0)
-_
0(0)
+
-yJ tan 01
:--
0(0)
+
-yV/p2
+
q2
.
(6.3)
This shows that the surface free energy is non-analytic (as a function of the slopes p and # The line tension -/ is a function of the step orientation
local
y"
-_
dy/dx
--
-p1q.
As
a
matter of
fact,
it is
an
analytic function
at all
singular behaviour of the surface free energy at low temperature is typical of the high-symmetry, low-index orientations, which are therefore T. The
singular. These appear as flat facets in the macroscopic equilibrium crystal. The linear size of a facet is proportional to -/. These shape statements are proved using the so-called Wulff construction, a geometrical construction which relies the surface free energy of 4 crystal to its equilibrium shape [3]. Using the projected free energy 0 is useful to derive the analytic equivalent of Wulff’s contruction from a variational principle: the equilibrium shape of a crystal minimises the surface free energy for a given volume [4,3]. called
of the
6.1.4
Surface Stiffness
equation (6.3) may hold at all temperatures. Steps can 0. At finite temperature, deriving (6.2), only at T they must fluctuate-in what is called step meandering. It may well be that a step becomes so "delocalised" that the concept of step looses its meaning altogether, above a given temperature. Indeed, we will see in Sect. 6.2 that such a temperature exists, and the corresponding transition is the roughening transition; in the rough, high-temperature phase, the step line tension ^/ actually vanishes. What is the form of the surface free energy at high temperature? Our experience with critical phenomena suggests that it may be useful to assume for the free energy an analytic expansion a la Landau in the disordered, high temperature phase [4]. To do that, it is convenient to transform the surface integral in (6.1) into an integral over Cartesian coordinates x and y. For a closed surface, this requires that the surface be cut into a few pieces and different projection planes be used for the various pieces. Locally, It seems doubtful that
be
straight
as
assumed in
--
Alberto
202
i. e. for
Pimpinelli
given piece, it is convenient 0 of equation (6.3),
energy
0(p, q) so
that
(6.1)
can
be rewritten
F
Take
to introduce the
a
reference the
now as
--
=
a(p, q)
as
follows:
+
surface free
q2
I 0(p, q)dxdy
F0 +
plane
11 + p2
projected
0.
z
(6.4)
Expanding 0
and q, and noting that the linear terms in p and q well as the mixed second derivatives 0’12 = 0’21 =
to
can
quadratic order in p vanish, as
be made to
02 alOpOq, (6. 1)
and
(6.3)
become F
where o-1
(within
--
an
_-
Fo +
Oulap,
0.11
1 2
11 dxdy [(0.
=
we
0.11) P2 +(0- + 0’22) q2]
a20.1OP2, etc. Letting constant):
0"11
-
0’22
-
0’//,
one
obtains
immaterial additive
-_I& (P2+ q 2)
(p, q) where
+
introduced the
(6.5)
2
surface stiffness &
=
0-
+ 0-"
.
obtaining (6.4) we started from an analytic surface free energy. In other words, we treated the solid as a continuum, completely disregarding the discrete structure of the crystalline lattice. Does it matter? The answer is no, above the roughening temperature, as we argue in Sect. 6.2. For
6.1.5
Surface Chemical Potential
The chemical
potential
is the Gibbs free energy per
particle (here,
per
atom),
and it is defined from y (T,
(ON) OG
P)
OF
y(T, V)
ON
The chemical
(6.6a) T,P
)T,V
(6.6b)
potential determines, at equilibrium, the shape of a crystal. Indeed, it is also useful to describe the time evolution of a surface not too far from equilibrium. At equilibrium, the chemical potential of an infinite system must be constant everywhere (if more than one kind of particle are present, the chemical potential of each species must be constant): Y
-
Yo
Statistical
6
Physics
Crystal Surfaces
at
203
finite size system such as a crystal of fixed volume V? Let us locally modify the crystal surface so as to change isothermally and at constant volume the number of particles by a quantity How is this relation. modified for
a
appropriate thermodynamic potential to use is the Helmoltz yJN. Introducing the atomic Then, p is found from (JF)T,V
JN. The most free energy. volume v --
=
VIN,
let
z(x, y)dxdy
N V
If
z(x, y)
then
by Jz,
varies
.
(JF)T,V
y(T, V)JN
--
=
V
if Jz(x, y)dxdy.
(6.7)
Varying (6.4) yields: JF
10 =
V
if
Jz (x,
y) dxdy
Sp I I ( LO Op
+
integral
The term within brackets in the second
provided 0
c9p
C9X We
Sp) Op
=
Jz,
can
--
if JZ(X’ Y) I
Finally, equating Mullins
the
following we
volume
integrated by parts,
so
(
a
Ox
00
dxdy.
OP
that the first term vanishes. The
(00)
-
OX
preceding equation
to
( )] 00
’9
aq
ay
Op
(6.7)
same
one
way, and
we
dxdy.’
gets (Herring 1953,
1963): Y
In the
(6.8)
c9z/Ox,
=
be handled in the
Ito V
JZ
-
find: JF
dxdy.
0 (Jz)) dxdy I f (L Op
dxdy
(6.8)
be
)
OX
local variations
only consider
second term in brackets in
Jq
’9
dxdy
( 0 JZ)
Oq
can
is at least twice differentiable. Since p
00
00
+
v
potential potential
--
at the
P0
-
[c9x (2p)
V
will often choose
our
+
C9 Y
(6-9)
c9 q
units in such
a
way that the molecular
potential po is the chemical bulk solid, according to (6.1). At equilibrium, the chemical surface must also be equal to yo, and from (6.9) the surface
VIN
of the
=
is 1. The reference chemical
must be flat.
If
(6.4) holds,
the chemical
potential
(6.9), given by 61-t
is P +
(92Z
a2Z
=
-V&( TX-2
Jp,
+
9y2
where
Jp is, according
to
(6-10)
Alberto
204
Pimpinelli
where & is the surface stiffness. In’the &
=
o-.
Locally,
surface
convex
any
Z(X’ Y) where R, and R2
Inserting (6. 11)
are
into
the two
(6. 10)
particular,
for
a
sphere
of
case
(x. y) I
(x2
-
2
an
may be
R,
+
isotropic surface tension, approximated as follows
Y2
(6.11)
R2
principal radii
of curvature of the surface.
find the Gibbs-Thomson relation
we
61t In
ZO
z
=
V&
(Ri 1
1
+
(6.12)
R2
of radius R 2vo-
R A similar
expression yields the
excess
chemical
potential
in the
vicinity
curved step on the surface, as a function of the step stiffness = d27/dO2’ where ^/ is the step line tension. Expliciting the atomic area
letting
Note the factor 2 in
6.2 6.2.1
-Y(O) a
2 ,
R be the local radius of curvature of the step, the relation reads
Jy
(6.13)-a
Morphology
of
Adatoms, Steps
a
=
a
of
2
a
+
and ,
(6.14)
*
R
consequence of the 2 terms in
(6.9).
Crystal Surface
and Thermal
Roughness
of
a
Surface
a high-symmetry surface-e.g. (100) or (I I 1)-at equiperfectly flat, i.e. it should contain no step. At low temperature, there are a few free atoms, or adatoms, and vacancies. Their density depends on the energy one has to pay to create an adatom-and, as a consequence, a vacancy-from a terrace. In the approximation where bond energies are additive (broken bond approximation), and only nearest-neighbour interactions are considered, this energy is equal to 4E on an fcc-(001) surface and to 60 on a (111) surface, if E is the bond energy. At higher temperature, clusters of atoms start to appear. Such clusters are closed terraces bounded by steps, at equilibrium with a two-dimensional gas of adatoms. Indeed, adatoms are continuously absorbed at and released from steps. Atoms emission from steps requires an energy Wad) much smaller than the energy cost for extracting an atom from a terrace-indeed, on an fcc surface the former is just half of the latter. Then, we expect the equilibrium adatom density neq to be given by a Gibbs formula
At
zero
temperature,
librium should be
neq
--
exp,
(-#Wad)
1
(6.15)
where
1/#
=
Physics
Statistical
6
k.8T and kB is the Boltzmann
at
Crystal Surfaces
205
constant.
The step density (total step length per unit surface) increases with temperature. This increase is not easily seen directly by microscopy. Indeed, most of
microscopy techniques work best
at low
temperature, where
matter
transport
equilibrium. On the easily other hand, the atom scattering or x-ray diffraction signal does exhibit a dramatic change when temperature is increased and the surface roughens. Above some temperature, the lineshape, which is lorentzian at low temperature, undergoes a qualitative change. One can for instance measure the "specular" reflection, i.e. that whose reflection angle almost equals the incidence angle (speculum is the Latin for mirror). The specular peak (as well as the Bragg peaks) is narrow for a smooth surface while a rough surface scatters radiation attain thermal
is hard and the surface does not
in all directions.
The
[5-7]
interpretation
patterns from
of diffraction
a
hot surface is difficult
roughness to broaden clear, from a quantitative analysis, that the greatly increased by heating.
because the effect of atomic vibrations adds to
the reflected beam. It is however total step
The
is
length
reason
is
essentially the following.
At
temperature, creating a W, does not change much,
zero
an energy (per length) the step entropy increases, so that the free energy decreases. A simple estimate is possible, if we consider on the (001) face of a cubic crystal a zig-zag
step
W1. Even if
unit
costs
step, whose average direction makes
angle
an
of ’45’ with the bond directions.
configurations of a possible approximate each left to from walk random step of the random right, going square-lattice walker being parallel to a lattice bond, and backward steps being forbidden. If the width of the system in the direction of the walk is uniformly equal to the bond length multiplied by Lv _2, all configurations have the same energy An
if
calculation is
2LW,. Since there
are
2 2L
we
configurations,
consider the
the entropy is 2L In 2 and the free
energy per bond is
W, The free energy per unit length Since a, the lattice parameter, is
-y1a
(6.16)
kBTln2.
-
is called the line tension chosen
generally
lecture notes, the term line tension is often
as
employed
the
length
of steps.
unit in this
for 7 itself.
When -y is positive, one has to provide mechanical work to introduce astep into the surface. If the total step free energy L7 vanishes, steps can that appear spontaneously-they cost nothing. Thus, equation (6.16) tells us the surface
undergoes
a
transition at
a
temperature TR approximately given
by TR
This transition is called the
W, -
kB In 2
roughening transition.
(6.17)
Alberto
206
6.2.2
The
As
above,
seen
Pimpinelli
Roughening the
Transition
roughening
transition temperature may be defined as the steps vanishes. According to the
at which the line tension of
temperature
experiment [8]
the line tension does vanish at
some
temperature, and its
behaviour agrees with (6.16). Near TR, the experimental curve bends away from the straight line predicted by formula (6.16): one can wonder whether it is
instrumental effect
an
section, it is
fundamental
or a
As will be
one.
fundamental effect. But before
a
seen
discussing that,
we
in the next would like
to make three remarks.
1. The
*
entation,
roughening
transition temperature
i.e. it is different for
a
(111)
and for
a
depends
(1,1,19)
on
the surface ori-
surface. This
point
will be addressed in Sect. 6.2.6. 2. For
*
given
a
orientation,
the step line tension vanishes at the
3. Formula
ative for T >
so
(Problem: prove this.) (6.16) suggests that the step
step orientation. *
surface
temperature for all step orientations,
same
TR. Actually, it is
not so, if
that Tp is
independent
of the
line tension may become negstep is defined in a model-
a
independent way. (Problem: could you think of the appropriate definition?) Indeed, the concept of a step looses its meaning above TR.
Smooth and
6.2.3
The
roughening
Rough
Surfaces
transition is much
more complicated than suggested by the Indeed, thermally excited steps are not isolated objects, they are closed loops. In this Section, we shall try to give an idea of what a rough surface really is. The reader will find more details about the roughening transition in the monographies by Balibar & Castaing (1985) [9], Van Beijeren & Nolden (1987) [10], Lapujoulade (1994) [11], Nozi6res (1991)
discussion of Sect. 6.1.2.
[12]
and Weeks As
seen
(1980) [13].
in the
previous section,
TR. It is therefore appropriate way. Consider
the concept of step is not useful above
to characterize
roughness
in
an
alternative
infinite surface of average orientation perpendicular to the z axis. Let (x, y, z) be a point of the surface. The "height" z will be assumed to be a one-valued function of x and y, so that "overhangs" are excluded. Let r1l
--
an
(x, y)
height-height
be
a
point of the two-dimensional (x, y)
g(R11)
-=
( [z(rll)
-
The interest of this function is that it
tatively
space. We define the
correlation function
z(r11
has,
+
if
R11 )]2
gravity
(6-18) is
neglected,
different behaviours above and below TR:
lim
JR111-+oo
g
(I R11 1)
finite for T
TR
oo
at low
Physics
temperature, in agreement with
(6.19a),
low-temperature expansion [3]. It is more difficult to and only a plausibility argument will be given. It is indeed prove (6.19b), reasonable to assume that, at sufficiently high tempqrature, say kBT > W1, the discreteness of the crystal lattice becomes negligible, so that the surface height z(x, y) may be regarded as a differentiable function of x and y, as it is for a liquid. Thus, we just forget that we have a crystal, and we write the can
be
proved by
a
surface energy as if it were a continuous medium. The surface free energy F ,,,,f is then simply proportional to the surface area, and the proportionality
(cf. 6.5). Introducing
surface stiffness
coefficient & is the
the local
slopes
p
and q,
F,u If
dxdyVF17+i-p
6
(thermal fluctuations)
or, for small undulations
FE;u,f
-
Co.nst +
=
dxdy
2
+q2
of the
surface,
(P 2
2)
+ q
(6.20a)
However, in our world subject to gravity, thermal fluctuations of the surface acquire an additional energy from gravity. The effect of gravity is irrelevant below TR (Problem. Why?). It is not so above the transition temperature. The energy of a column of matter of cross section dxdy, whose height is between z, and z, is
pgdxdy
f
Z
(d(
=
pgdxdy(Z2.
_
Z2I )/2
’.
where p is the specific mass and g the gravity acceleration. The term containing z, is constant and will be omitted. The energy excess associated with surface
shape fluctuations
and
resulting
from both
is
F
2JJ dxdy 1
(6.20b) q2)] Y
by using
g(R11)
the
-
2
7.
(19Z2 -
C9 X
(9Z +
ay
and surface tension
2)
(6.20b)
readily obtained by Fourier transforming 2 kBTI[pg+ &(q + equipartition theorem (I Z2q 1)
(6.18)
The correlation function and
pgz
gravity
is
=
2kBT
jila dq,dqy n 0
cos(q R11) + &(q2 + q2) Y
X
-
P9
(6.21)
X
approximation of g(R) may be obtained if the lower limit of integration is replaced by an appropriate cutoff, below which the numerator is almost zero. This allows us to replace the cosine by its average value 0, and one obtains An
g(R11)
_-
27r
kBT &
In
(
pg +
2
&/a &/ JR11 12
pg +
(6.22)
Alberto
208
If
gravity
goes
Pimpinelli
be
can
which is
neglected,
possible
JR11 I
for
< A
=
V1o_-1pg, (6.22)
as:
g(Rll)=Const.+(47rkBT/ ,-)I-nlR,llI This proves function
(6.19b).
A surface is called
g(R11) diverges
Previously,
as
(6.23),
in
defined the
we
(6.23)
rough if the height-height correlation no divergence.
and smooth if there is
roughening
transition in terms of the vanish-
ing of the step free energy 7. We should worry about the equivalence of the two definitions. Fortunately, they are equivalent. (Problem. Prove this equivalence.
proof of (6.23) relies on the use of (6.20b). If the step positive, equation (6.20b) cannot be true). Indeed, if -/ > 0,
Hint: the
line tension is
the formation of
terrace of size L has
a free energy cost which diverges free forbidden means diverging energy configurations-i.e., whose statistical weight vanishes-and the correlation function g(R11) is finite for a
with L. A
JR11 I
Therefore, the roughening transition can be defined (at least on high-symmetry surface) either by (6.19a,b) or by the vanishing of ^/. The role of gravity, which kills the roughening transition, becomes negc Ipg. The ligible at lengthscales shorter than the capillary length A order of magnitude of & is typically the energy of a chemical bond, i.e. IeV per atom. The resulting value of A is a few centimetres, i.e. much larger than the distance over which equilibrium can be reached at a crystal surface. For this reason, gravity is usually neglected in surface science. -* oo.
a
--
Diffraction from
6.2.4
Another
a
atom
scattering
do-
A
dS2
q’
where A
h(X, Y) mal
cross
peculiarity of the roughening transia scattering experiment. The x-rays section has the form (see Chap. 2)
dXdY
_2
depends
on
describes the
averaging
over
Surface
consequence of the
important
tion is the form of diffraction or
Rough
peaks
e
-
i’l [h(X,Y)-h(O)]
the electronic
shape
in
density
e
_i(g"X+qYY)
(6.24)
of the surface atoms and
of the surface. The
angular brackets
z
--
denote ther-
surface fluctuations. In order to compute this average, it assumption that fluctuations have a Gaussian prob-
is customary to make the
ability distribution. With this assumption. it
is
straightforward
to show that
(Chap. 2)
r(R11) where
we
wrote
equation (6.25)
preceding
-=
Rjj we
(e_ iq ,[h(Rjj)-h(O)] ) =
(X, Y).
recognize
In the
the
section. The function
q2 -
e
2
exponent
height-height
r(R)
([h(Rjj)-h(O)]2)
takes the
at the
right-hand
(6.25) side of
correlation function of the name
of
pair correlation
Statistical
6
Physics
Crystal Surfaces
at
209
appreciate the relation between the height-height and the pair functions, the latter being the quantity directly measured in a scattering experiment, it is instructive to consider a few examples: i) g(R11) C. The pair correlation function is also a constant,
function.
To
correlation
=
r(R11) and the scattered
ii) g(R11)
intensity
CIR11 I.
=
The
’pair
Lorentzian in this
a
cross
cg2’ -_
us
e-.
z
iii)
g
(R11)
an
exponential
JR111
(6.26) intensity 1, which
2
+
4q211
the components of the momentum transfer wavevector
parallel to the average surface. pair correlation function has I R11 1.
respectively orthogonal -_
delta function.
to find the scattered
C2q4 are
a
case:
2Cq
where q, and q1I
is
correlation function is
section allows
scattering
is
,
(6.24)
obtained from
r(R11) The
C’
=
C In
and
The
a
power-law
behaviour
1’(RII) as
well
as
the scattered
=
IRIII-Cq,2
e- Cq -2InIR111
(6.27)
intensity 2
2+Cq.
q
The
case
of
a
vicinal
(stepped) crystal
surface will be treated here
[14].
At
peaks expected. g(R11) a constant, Beyond TR, the diffraction peaks are expected to acquire a power-law shape. However, a vicinal surface has two non equivalent directions, parallel and orthogonal to the steps. When computing the scattered intensity we must integrate only along a vector orthogonal to the steps. Then one finds and delta-function
is
low temperature
I+Cq 2,
q
At the
peaks
position behave
perature, C
of
as
--
I
largest sensitivity, -
q 11
+C7r2
/a2,
are
qz
=
7r/a,
where C is
the
intensity
actually
a
of the diffraction
function of the tem-
QT). According to (6.23), at TR the height-height correlation case (iii) above, and C(TR) is equal to the universal
function has the form of
value 2a 2/7r2
.
Thus,
the scattered
intensity
at the transition
temperature TR
reads q
and the
roughening temperature
can
d T
-_
d
1+2
be -found
(In 1)
(In q11)
as
the
temperature where
Alberto
210
Pimpinelli
Capillary
6.2.5
In the
of
case
an
Waves
isotropic surface, such as the surface of a liquid, (6.20b) surface, the so-called capillary waves. in Sect. 6.2.5, equation (6.22) implies that the reflected
describes thermal excitations of the
As
have
we
seen
of e.g. x-rays from the surface allows a direct measurement of the surface tension o- (which coincides with & for a liquid).
intensity
Surface Growth and Kinetic
6.3
Equilibrium
6.3.1
At any temperature, at pressure
crystal
a
is in
Psat. Assuming the
::::::
Vapour
equilibrium
with its saturated vapour
ideal gas with density Psat, rate of atoms onto the surface [4,1]:
vapour to be
theory dictates the impingement
kinetic
Rimp
with the Saturated
Roughening
an
Psat1’/2__7rMkBT-
that all the atoms are adsorbed-equivalently, that the sticking unity, which is not necessarily true with molecular speciesdetailed balance gives the equilibrium adatom density neq once the saturated
Assuming
coefficient is
vapour pressure is known. If Tev is the average lifetime of
evaporation,
then
Rimp
neq/Tev,
Revap
--
an
adatom before
and
neq
Psat
Tev
v ’27rmkBT
(6.28)
right-hand side of this equation is called the evaporation rate of the crystal. The saturated vapour pressure for a given solid is obtained from the equation of state of an ideal gas; the detailed derivation is in [1]. The result The
is
Psat
Inserting (6.29)
into
(k BT)
-
(6.28) neq
we
obtain the
M
’rev
( 27rh2 )2 exp(-PWcoh) M
2
47r2h3
evaporation
(6.29)
-
rate in the form
(6.30)
exp(-13Wcoh)
time-’]. Its exponential has dimension [length the to exponential, so that it is compared temperature dependence coefficient T -’. Numerical values of I/To usually written as a T-independent s-1 ’k-2 at for most elements, room temperature. are of order 1014 The
quantity
-2
in front of the
is weak
6.3.2
In the
Supersaturation equilibrium state, equal: 1-tsolid
must be
-
and Vapour
the chemical Yvapour
=
Deposition
potential
of the solid and of its vapour we increase the
Peq- What happens if
Statistical
6
Physics
at
Crystal Surfaces
potential in the vapour? The vapour atoms go potential is lower, i.e. to the solid. The solid grows! The chemical
/-1vapour
-
-
211
where the chemical difference
Yeq
supersaturation. The chemical potential is not easily controlled experimentalist. It is easier to change the pressure. For a vapour
is called the
by
the
treated
as an
ideal gas, the pressure P
Therefore,
for
0Ap
---::
can
Psat
rate i of
growth
as
exp (OzAp)
1,
tcrossover)
of the
of the surface width with the
system size L: Wsaturation *
P
is the
growth exponent, which characterizes the increase of the
surface width with the time t
(at
z
is the
tcrossover):
t
geometry,
q., and q,
are
addition, sample area is illuminated during the scan. 0. In this geometry, the reciprocal space is measured 0 scan at fixed w in the qy-direction normal to the incidence plane. This type of measurement is generally less used than the others because most reflectometers have no -motion out of the incidence plane even though it offers a full qy range accessibility [4]. It is usually preferred to let slits widely opened in the y-direction leading to an effective integration over qy. -
no
=
as soon as
w.
In
a
constant
7
Experiments
on
Solid Surfaces
219
0.25
0.20
015
2
r
o
o
specular reflectivity
longitudinal diffuse scan
oooc’oo oo
/
detector
A
00=o.1,
ooooo rocking scan
scan
0 9*
/20=1.5"
0.10
0.05
rinaccessible q-
0.00 -2.0
-1.5
inaccessible q-area
area
-1.0
-0.5
0.0
1.0
0.5
1.5
2.0
qX(j 0-3A-1) Scans in
Fig. 7.2. The line
scans are
reciprocal
space with
of the inaccessible
representation
areas.
shown for the CuKal radiation
Resolution The choice of slit widths or monochromator and analyser crystals is essential for setting the resolution of the measurements for given configurations [3,5-7]. The calculations of different resolution functions can be
Considering the case of in-plane scattering scattering angles, one gets from Eqs. (7.1):
found in the literature.
with small
I
Jq, Jq,
kO(Oj2nzAOj2n + 02 ko(ZAO + //\02 in
SC
in
A02
0)
(7.2)
SC
SC
in
monochromatic radiation, and incoming and outgoing beams with anguOsc -- 0 and acceptance zAO,,c respectively. Assuming Oin
for
a
lar
divergence AOin
and ZAOin
;z
zA0,;,
;: -,
zAO,
one can
estimate the resolutions
Jq, q_,
q ,z_AO
as
(7.3)
2kOAO.
experimental results reported in the following section from meaon a triple axis spectrometer using flat Ge(111) monochromator and analyser and the CuK, radiation, the typical values for resolution are 2.10-3A-1 and Jq,, 2.10-’q,. Jq, The resolution functions determine the maximum length scales which can be coherently proben by the experimental measurements. For the above men107A. 10’A and Xinax ,: 104 / q, tionned resolution, they are Zmax
For the
surements
,
220
Jean-Marc
Gay
and Laurent
Lapena
Data
Analysis Before any comparison with theoretical simulations of reflectivity, the data may have to be corrected for various geometrical factors depending on the measurement, configuration [6,7]. Alternatively, the simulations may include the corrections and the data are considered as they are. The specular reflectivity is defined (see Eqs. (3.5), (3.6)) as the ratio of the reflected intensity at the scattering angle 20 to the intensity of the direct through beam. For very grazing incidence angles Oin, the sample surface (length I along the x-direction) is almost parallel to the beam so that a fraction of the incoming beam (width bin) does not illuminate the surface and cannot be consequently reflected. This geometrical effect is responsible for the bump experimentally observed at low angles. A plateau of total external reflection below the critical angle, as theoretically expected, is obtained upon renormalisation of the experimental data. Assuming a uniform rectangular flux distribution of the primary beam, the correction factor is simply expressed as a function of Oj0,I arcsin(bin/1) by: =
in
fsp ec, in fspec,in On the detector
side, slits
=
Sill Oin / sin
=
I
or
and
Os,
is
in
for for
Oin Oin
bin
bc
Ofn. For the data normalisation, one has to consider a correction factor proportional to the actual illuminated area: sured diffuse
in
fdiff,in (Oin) hiff, in (Oin) This is the normalisation
for Oin
j(qll)
=
[d(qll )]2
substrate)
1
and of the
Okk(qjj) [&(qjj)](1’_j)
(8.15) inter-plane
(8.16)
physical meaning of the particular terms in (8.15) is obvious. The first on the right hand side represents the influence of the substrate surface modified by the replication function) the second term is due to the intrinsic roughness of the layers beneath the layer j. Knowing Ojj(qll) we can calculate the mean square roughness O-J2 of the jth interface: The
term
O_j2 8.3
=
(Zj2(r,,))
dqjj Ojj(qll)
(8,.17)
Setup of X-Ray Reflectivity Experiments
experimental setup to investigate the fine strucintensity pattern in vicinity of the origin of reciprocal and exit with respect space (000) under conditions of small angles of incidence to the sample surface.
In this section
we
outline the
ture of the reflected
Tilo Baumbach and Petr Miki-ilik
238
Experimental Setup
8.3.1
A conventional x-ray reflectometer is drawn in Fig. 8.3. The x-ray source (a conventional x-ray tube or a synchrotron) emits a more or less divergent and
polychromatic beam. The -nonochronuytvr i1a crystal or a multilayer mirror) and entrance slits produce a sufficiently monochromatic and parallel beam, hitting the sample surface under the incident angle Oil,. Its angular divergence is characterised by the spatial angle AQin. The sample is mounted on a goniometer, which allows one to change the incident angle Oin by the rotation w. The x-rays are reflected (scattered) by the sample. The coherently reflected beam leaves the sample in specular direction (under the exit (final) angle 0,,c 0,,c in the plane of incidence). Due to roughness there occurs diffuse scattering into the upper half space of the sample. A detector rotates around the sample and measures the Pux of photons (in units of counts per second) through the detector window, which defines the spatial angle interval AQdt around a certain spatial angle S?.,c (sufficiently defined by Os, in the coplanar case). If we suppose a perfectly monochromatic and parallel incident beam of intensity -10 then the idealised flux through the detector window is related with the differential scattering cross section by =
J
Taking
the
account,
i
=
divergence
we
JAd?j.
Actually,
in
lo
I
0
do-
-_
+AQdet/2
lo
and the
(dQ
intensity profile
do-
dQ
(8.18)
.
of the incident beam into
obtain d2
d,/-A Qj 1, lo (A Qjj
of
+Afldet/2
)Jf2,_,-cAd2det/2
large sample,
do-(S2jj +zAQj, Q) dQ
dQ
.
(8.19)
the detector slits select another
angular sample area. That can be overcome replacing the detector slits by an analyser (also a perfect crystal or a multilayer mirror) in front of the detector similar a triple-crystal diffractometer (TCD). The monochromator is the "first crystal", the sample the "second case
interval for each
a
point
on
the illuminated
Monochromator
Analyser
A- ,‘D&tectoir
ar
Sample
Fig. 8.3. Schematic setup of an x-ray reflectometer (source, monochromator, sample, slits and detector) and of a triple-crystal-like diffracton-leter (source, monochromator, sample, analyser and detector)
X-ray Reflectivity by Rough Multilayers
8
crystal"
and the
analyser
the "third
crystal".
The flux measured
by
239
the TCD
is
L where
d A Qin 10 (ZA f2in) i.
D(zAS2)
8.3.2
is the
f
dS2
do-(0i, +, Af2in, 0) dS2
V (S2
-
Qs, )
(8.20)
,
reflectivity of the analyser.
Experimental
Scans
Mapping the measured flux for different angles of incidence and exit we can or by plot the measured scattering pattern in angular space, J(f2in, three reciprocal space coordinates and one angular coordinate of the sample, kin, Vin). Restricting ourselves on coplanar reflection (ks, kin e.g. j (k,;, and the surface normal are in the same plane), the angular representation J(Oin, 0,;,) and the reciprocal space representation J(q) with the scattering vector q kin are equivalent. k 3, The principal rotations of a (coplanar) TCD are: -
=
-
arrangement in the coplanar scattering measures the scattering angle (20 Oi,,+O,,,), plane around the sample: the variation of 20 changes 0,,, (A20 AO,c). The rotation w of the samp le around the same axis: Oin 20-w, a w, O , variation of w changes simultaneously Os, and Oin (ZAW Oin -AOsc)
1. The rotation 20 of the detector
20
--
2.
-
AWA20
*
Q.
Fig.
Q11
QZ
experimental scans in the reciprocal space. Right figure enlargement around its origin, where x-ray reflection takes place. The
8.4. Illustration of the
shows the 20-scan
-
(detector-scan)
represents rocking with qjj
=
0
follows the Ewald circle of the incident
scan, which is transversal for XRR. For 2W
(speculax scan)
wave.
=
The
20 it is
a
W-scan
q-,-scan
Tilo Baumbach and Petr Miktdik
240
Different
experimental
Fig. They are:
detector
be
scans can
8.4 the most usual
In
scans are
performed by coupling both rotations. reciprocal space.
illustrated in real and
20-scan. The incident wave vector kij opens out the Ewald keep the angle of incidence fixed (w const) and rotate the detector arrangement, we move in reciprocal space along the Ewald sphere c. w-scan or constant q-scan. The w-scan rotates the Ewald sphere around the origin of reciprocal space. Fixing the scattering angle 20, we fix the modulus of the scattering vector. Then the w-scan represents a constant q-scan since we move in reciprocal space on a circle of radius q q around the origin. zAO/A20-scan or radial scans. Rotating the sample and the detector arrangement in a ratio zAw/zA20 1/2, we drive the TCD in reciprocal space in radial direction from the origin of reciprocal space. 0/20-scan on the q,, axis or specular scan. This special radial scan with 0., and performs a q,-scan at w120 1/2 keeps the condition Ojj 0. This experimental mode is also called specular scan, since the q., detector selects always the specularly reflected beam. q.,-scan and q,,-scan. These scans go parallel to the q., and q,, axes at fixed q,, and q., position, respectively. scan or
sphere
c.
If
we
=
--
--
-
=
Sometimes it is useful to
reciprocal space map, i.e., to meaintensity by combining different scans, e.g.
measure a
the map of the scattered measuring a series of w-scans sure
(rocking-scans)
in the interval from W--O to
varying position-sensitive detetor (PSD), one would Using for different PSD-spectra omega positions. The angular region investigated by a reflection experiment is limited by the horizon of the sample. The limiting cases for grazing incidence (0ij 0) and grazing exit (0, 0) are illustrated in Fig. 8.5. The situation in is reciprocal space represented by the two limiting half spheres co and ’E". X-ray reflection experiments are usually realised at very small scattering angles. In Fig. 8.4(right) we show the introduced experimental scans in the x-ray reflection mode and their restrictions due to the sample horizon. Especially the w-scans are narrowed down. In the accessible region of reflection, w--20 for
20.
a
detect
-_
--
Grazing
incidence
Grazing
exit
OK.
K
2n& 4./)L
Fig. 8.5.
Situation of
and real space
grazing
incidence
(left)
and
grazing
exit
(right)
in
reciprocal
X-ray Reflectivity by Rough Multilayers
8
i.e.
near
the
transversal
of the
origin
reciprocal
space,
241
they perform approximately
a
(qll-scan).
scan
Specular X-Ray Reflection
8.4
In this section
we
discuss
experimental examples
theoretical and
some
of
structures with the aim to show
specular x-ray reflection by features created by different
layered surface. roughness point properties. The coherent scattering intensity is concentrated along the specular rod. That means, the appropriate experimental scan is the specular or 0/20-scan. coherent
typical
with
Roughness
8.4.1
a
Gaussian Interface Distribution Function
Single Surface The predominant number of samples have been successfully characterised assuming a Gaussian probability density of the interface roughness profile (see (2.19)) Pi
In this case,
(e.g.
a
(Z)
o-- ,/_27r
e---’ /2
or
2
(8.21)
Chap. 3, Eq. (3.103), we obtain for a single amplitude ratio of dynamic reflection [16,17]
shown in
as
substrate)
the
r,,h dyn with the
=
amplitude
flat
rdyn
::--:
e
-2k ,,Ok.,
,or
ratio of the flat substrate
2
dyn
see
(8.22)
being
reflection coefficient of the substrate surface rflat
surface
the
dynamical
Fresnel
(k;,,o-k_’,j)1(k,,o+k_’,j),
Eq. (3.68). The ratio of kinematical
reflection
rcoh kin
’::::: ’
coefficients is
flat
rkin
(Eq. (3.104))
2
-
0-
2k. ’00,2
with the kinematical Fresnel reflection coefficient of the surface
(8.23) rflat kin
=
214q 2
qC
Z;
Eq. (3.9 1). Both the kinematical and the
multiplied
with
a
dynamical Fresnel reflection coefficients are containing the r.m.s. roughness 0- in the
diminution factor
exponent. The kinematical diminution factor decreases with the
square of
proportional to the angle of incidence. Its scattering form resembles the static Debye-Waller factor. The dynamical diminution factor contains the product of the scattering vector in vacuum q,,o and that in the medium q,,,. The angular dependence of the diminution factors in the dynamical and the kinematical theory differs substantially for small angles near the critical angle of total external reflection 0, see Fig. 8.6. Neglecting absorption, the scattering vector q,,, becomes purely imaginary below 0,. Consequently there is no influence of roughness on the reflectivity in this the
vector q, which is
Tilo Baumbach and Petr Mikulik
242
101L
10,
10-1
1Oo 10-, 10
10 0 0.0
-2
0
0.1
0.2
0.3
OA
10‘
10, 10-4
10-4
107,
e
10"0!0* ’07 ’0:4’0.6 angle
Fig. 8.6.
0.8
1.0
of incidence
The coherent
1.2
1.4
1.6
1
7; .0
0.2
0.4
(deg]
0.6
angle
1.0
1.2
of incidence
0.8
[deg]
1.4
1.6
reflectivity
of a rough Si surface. In the left panel the refleccompared with that for the roughness a =I nm, calculated by the "dynamical" theory (8.22) (full) and the kinematical theory (8.23) (dotted). The kinematical reflectivity diverges at grazing incidence. The "dynamical" curve coincides nearly with that of the flat surface below the critical angle 0,-. In the subfigure, the dashed line represents the coherent reflectivity of a rough surface calculated with dynamical Fresnel reflection coefficient and kinematical diminution factor. Thus the reflectivity decreases also,below 0,. In the right figure, influence of different roughness, calculated by dynamical formulae, is demonstrated. Close to 0,_ (see subfigure), no essential change is observed
tivity of
a
flat surface
(dashed)
is
angular range within the dynamical diminution factors coincide. A
description.
At
detailed discussion of both formulae
large
incident
(8.22)
and
angles
(8.23)
both
is
given scattering specular direction has been studied by means of second order DWBA, showing its dependence on the lateral correlation length A. Concluding therefrom, the specularly reflected intensity can be described by the "dynamical" equation (8.22) for short A below I pm. For larger A the kinematical formula (8.23) becomes more appropriate. with r flat kin Surface roughness of numberless samples of amorphous, polycrystalline and mono-crystalline material systems has been studied by SXR. In Fig. 8.7 we plotted one experimental example, the reflectivity of a rough GaAsin
more
[13].
There the contribution of the incoherent
to the
substrate.
Multilayer Conventional SXR-simulation and fit programs are today based a multilayer model with independent r.m.s. roughness profiles of each interface supposing a Gaussian probability density. This leads to effective on
Fresnel reflection and transmission coefficients
rj,j+l
-
r
flat
3,3+1
e
and
tj,j+l
(Eq. 3.103): --
tjflat+,,(k,,j-k,,j+l )2 7
.1/2
3,3
(8.24) for each interface. The influence
according
on
the transmission function is
to the small difference in the vertical
scattering
vector
rather
small
components
X-ray Reflectivity by Rough Multilayers
8
243
10, 10
0 Z
r \\
0
O.S.
I Z=CY
Z=
0.0
\20 -40
(D
1074 10-,
-20
020 20
’0 (A) z
40 40J
(points) and (line) reflectivity curves 12 A [18]. of a GaAs substrate, a Measured
F ig. 8.7.
calculated
=
0.2
0.0
0.4
0.8
0.6
1.0
1.4
1.2
1.6
In the inset the the surface is
angle of incidence [deg]
mean
coverage of
plotted
layers. However, the interface reflection is exponentially diminished by roughness, creating a strong change in the interference pattern. The effect of interface roughness versus surface roughness is shown in Fig. 8.8. The surface roughness mainly decreases the specular intensity of the whole curve progressively with q, where the interface roughness gives rise to a progressive dampening of the interference fringes (thickness oscillations). However, locally the variation in the Fresnel coefficients can cause more pronounced oscillations, too. In Fig. 8.9 we plotted the experimental and simulated curves of a magnetic rare earth/transition metal multilayer (Cr/TbFe2/W on sapphire A1203), grown by laser ablation deposition. It shows a quite complicated non-regular interference pattern. A good agreement with the simulation was realised by considering a thin oxide film at the sample surface. of the
1 100.1
10-2 0.01
10-3 10
A,‘.’,A
-4
0.00,
10-5 10-1
14\ 0
0.5
1
1.5
angle
2
2.5
3
of incidence
0.0001 3.5
4
4.5
[deg]
5
0
0.25
0*5
angle
1.25 1 0.75 of incidence [deg)
1.5
specular reflectivity of a single layer (20 nm tungsten) r.m.s. roughness and diminution factors. (a) (sapphire) Dynamical diminution factor. F om the upper to the lower curve: without roughboth surface and interness, interface roughness 0.5nm, surface roughness 0.5nm, face roughnesses 0. 5 nm. Surface roughness yields a faster decay of the refiectivity, while interface roughness attenuates the peaks. (b) Different diminution factors. Surface roughness 1.2 nm and interface roughness 0.3 nm calculated for the kinematical "slow" roughness (lower curve), dynamical "rapid" roughness (middle curve),
Fig.
8.8. Calculation- of the
on a
substrate
and without
for different
roughness (upper curve)
Tilo Baumbach and Petr Mikulik
244
.
..........
100000
2
10000 t
A12 03
1000
100 0
0.5
1
1.5
2
angle of incidence [deg]
8.9. Measurement (points) and the fit (full curve) of the specular reflectivity of Cr/TbFe2 /W multilayer [19]. We determined the thicknesses (34.6 nm W, 4.8 nm TbFe2, 5 0.5 nm Cr, 3 nm oxidised Cr) and the roughnesses (0. 2 nm above sapphire, 2.0 nm W, 0.9 nm TbFe2, 2.2-nm Cr)
Fig.
a
Periodic Multilayer The main feature of the specular scans of a periodic multilayer are the multilayer Bragg peaks, giving evidence for the vertical
periodicity,
see
Fig.
8.10 and Sect. 8.A.2.
10000
10
(a)
1000
b)
100
10
0.1
0
!A ;14
0.1
0.01
0.01
0.001
0.001 0.0001
0.0001 0
0.25
of incidence
angle
Fig. 8.10. Specular for
a
reflection
[GaAs (13 nm)
strate, flat interfaces
curve)
0.75
0.5
1
0
[deg]
by
an
0.25
angle "ideal"
0.5 of incidence
0.75
[deg]
periodic multilayer-calculated
curves
(7 nm)] superlattice with 10 periods on a GaAs sub(no roughness). (a) Comparison of the dynamical theory (full AlAs
with the kinematical
theory. The kinematical multilayer Bragg peaks
cor-
positions (000) RLP. The curve diverges at low incident angles. The dynamical calculation shows the plateau of total external reflection below the critical angle. Due to refraction the multilayer Bragg peaks are shifted to larger angles. The first multilayer Bragg peak broadening is caused by multiple reflection (extinction effect). (b) Comparison of the dynamical theory with the semi-dynamical approximation (single-reflection approximation [18]). The satellite positions of all Bragg peaks coincide, also the shape and intensities except for the intense Bragg peaks respond
to the
of the satellites of the
X-ray Reflectivity by Rough Multilayers
8
245
intensity ratio of the Bragg peaks depends on the layer set-up within multilayer period. The difference in the electron density determines the Fresnel coefficients, and the thickness ratio of the layers characterises the phase relations of the reflected waves of different interfaces. The laterally averaged gradual interface profile caused by interdiffusion or interface roughness leads to a damping mainly of the multilayer Bragg-peaks progressively with q, whereas the roughness of the sample surface reduces the intensity of the whole curve. This is demonstrated in Fig. 8.11. The
the
roughness roughnesses surface roughness no
10-1 10 t5 A
interface
----------
------
2
10-3 10"4 10-5
1
1
1
1
1
1
0
0.2
0.4
0.6
0.8
1
angle
of incidence
1 V 3 H9
1.2
1.6
1.4
[deg]
Fig. 8.11. Simulation of coherent reflectivity of a [GaAs (7nm) / AlAs (15 nm)]10x periodic multilayer with no roughness (full curve) or I nm roughness of surface
(dashed
lower
curve)
or
of all interfaces
(dotted)
Fig. 8.12 we plotted the measured SXR curves of an epitaxial CdTe superlattice on a CdZnTe substrate. Due to the low contrast of the electron density of both layer materials the first order Bragg peak appears only as a very weak hump on the slope of the surface. The other Bragg peaks have a shape similar to a resonance line. From the best fit we obtain the mean compositional profile. In
CdMnTe
decreasing roughness in multilayers The influence or decreasing during the growth from the substrate increasing roughness towards the surface can be described. by use of the roughness replication
Increasing
and
of
model introduced in Sect. 8.2. We start the ness
layer growth from
a
substrate with
a
Gaussian surface
rough-
profile, CNN (’rjj
For the non-random
-
replication
2
(8.25)
AN
ON
’rjj
function in
(8.10)
we
choose for all
layers
a
Gaussian function
a(ril
-
r1l’)
1,11 _11 112
1 =
-
7r L2
C
2L2
(8.26)
Tilo Baumbach and Petr Mikulik
246
10-
0.75
Ca) a)
0.5
10
>’
0
10-
PL re Pure
0.25
> C.)
LC(djTe
Pure
-2
3
T CdTe
4
0
8
12
16
position [nm]
z
10-4 10-5
10-6 0
0.25
0.5
1.5
1.75
2
[deg]
8.12. Measured and calculated
CdMnTe is
1.25
of incidence
angle Fig.
1
0.75
specular reflectivity of a [CdTe (14.2 nm) CdZnTe [20]. In the subfigure, the roughness effective MnTe concentration depth proffle
(2.5 nm)] 20 x superlattice
represented by
an
on
The factor L determines the loss of memory from interface to interface. This choice arises from the aim to explain the different limiting cases of roughness
by one class of functions. It is not supported by any physHowever, the model allowed to describe measured curves of SXR and NSXR showing good agreement [15,8].
replication ical
models
reason.
-
We
the intrinsic correlation function
assume
(8.13) ’
K(’rjj _,r111) Now
we
1,11 -11 ) (’ Ao.)2 e ( I
of all interfaces
2
(8.27)
-
AA
-
continue like in Sect. 8.2. The Fourier transform of the
correlation
function
is under these
N-1
Z2 1
Cjj (q)
-
(ONAN 2
2
in-plane
assumptions +
2
(Ao-zAA )2 2
(gAk )2 e
4
(8.28)
k=j
where
The
we
have denoted
A’
j
VA2
inter-plane correlation
is then
IV
Ci j, k (q) We obtain the
mean
?
O’j
square
=
+ 4L2 (N
j)
(8.29)
simply given by
Cjj (q)
roughness
dq Cjj (q)
-
2
ON
(qL)2 -
2
e
of the
AN2 W.:
i-k
(8-30)
jth interface N-1
+
(AozAA )2
E k=j
I
A k’2
(8-31)
X-ray Reflectivity by Rough Multilayers
8
VIA
0.1
(a) 0.1
2:1
0,01
0.01
.5
0.00,
0 .001
10-4
10-4
10-5
10 1
0.5
0
angle
Fig.
247
2
1.5
of incidence
-5
0
3
2.5
0.25
0.5
2
1 1.25 1.5 1.75 of incidence [deg],
0.75
angle
[deg)
(points) and simulation (full curve) of the specular reflecperiodic Nb/Si multilayer’ of 10 periods [19]. (a) Sample A, fitted by of constant roughness, (b) sample 13, fitted by the model of increasing
8.13. Measurement
tivity
of
a
the model
roughness
Let
us see
what does it
for
give
some
limiting
cases
of the model:
interface roughness is achieved with maximum replication and 0. Consequently o-j 0 and AoON, and all no intrinsic roughness: L substrate of the the ZN (X) surface, Zj (X) interfaces reproduce profile maximum obtained is surface the free towards by Increasing roughness o- > 0). From 0 and replication and a non-zero intrinsic roughness (L
1. Identical
=
--
=
2.
(8.28)
and
(8.31)
we
find 2
O’j
3.
-
-
2
2
ON +
(N
-
(8.32)
j)
describing the roughening during the growth. 0) leads Partial replication and no intrinsic roughness (L > 0 and zAosurface free the towards (smoothing of the to decreasing r.m.s. roughness described by multilayer during growth), --
?
17j
0’ -
2
N
-
1+4 (_L_) AN
2(N
(8.33) -
j)
replication occurs for diverging L, where a(,rll roughness profile of each interface is independent.
4. No
-
r1l’)
goes to
zero.
The
experimental example of two periodic Si1Nb rnultilayers, studies. The multilayer is grown by magnetosputtering for superconductivity thick with a Si02 layer and an Al buffer layer. deposited on a Si substrate buffer The roughness of the layer depends on its thickness and influences the
We compare here the
quality
of the interfaces. Two
investigated
and the results
samples
are
of different Al thickness have been
shown in
Fig.
8.13. The
multilayer period-
icity generates the multilayer Bragg peaks or reflection satellites, which are dampened by interface roughness. The roughness of the substrate and the
Tilo Baumbach and Petr Mikulik
248
buffer
layers has less influence on the reflection pattern. Sample A can be by a roughness model of constant r.m.s. roughness for all interfaces. The peak widths of the first intense Bragg peak is broadened by extinction due to dynamical multiple scattering. For all higher order Bragg peaks we observe a narrower (kinematical) peak width. The satellite reflections of sample B are also rapidly damped, indicating a large interface roughness. Besides the widths of the peaks increases with q.,. That can not be explained by model 1. The satellite intensities and shape can be successfully reproduced by supposing increasing roughness according to (8.32). Due to their increased roughness, the upper layers near the surface contribute with decreasing effective Fresnel coefficients to ’the reflected wave. Within the Bragg position the contributions of all interfaces are still in phase, however, slightly away from the Bragg condition the contribution of interfaces near the substrate and those near the sample surface do not cancel completely, giving rise to the peak broadening. fitted
Stepped Surfaces
8.4.2
The surface
morphology of monocrystalline samples can also be described by a discrete surface probability distribution following .LIj- L_j L__J .............X Z. the concept of terraces or small separated I islands. In the simplest case, the two-level 8.14. Multilayer with random Fig. consists of surface randomly placed islands two-level islands of uniform height d, so that the displacement z (r1j) has two possible values z, and Z2 d + z, with the corresponding 1 probabilities pi and P2 pi, see Fig. 8.14 [21]. The surface probability ..
I..dn.................n
F
.............
..
........
........................
...........
--
--
distribution function
p(z)
-
for this
P(Z) Since
(z (,Pll)-)
=
0, then Zi
-
:---
case
writes
P16(Zl)
-P2 d and
+
Z2
P26(Z2) =
(8-34)
pi d. The
mean
square
roughness
is 0’
2
=
p1Z21
and the characteristic function
X(q ,)
-
+ P2
(2.10)
e-iq
dP2
Z22
-
2
(8-35)
d)
(8.36)
PlP2 d
is
(PI
+ P2
e
iq,,
Putting this in the formulae for the reflected amplitude ratio of rough faces, we get the amplitude ratio of kinematical specular reflection rc9h kin A surface
forming
an
=
e-iq ,dp2
region perturbed in
upper and
a
(pi ro, 1
+ P2 ri 2e
this way acts
iq d)
as a
sur-
(8.37)
thin, homogeneous layer
lower interface with the Fresnel reflection coefficients
8
1
0o
..
...
.
X-ray Reflectivity by Rough Multilayers
...
10, V
:
...
111.11,11.1
........ ......
-two level surface -------
0 .............
10
gaussian surface
10
fiat surface 2:
t
(D
10-,
=
............
10
W
pl=0.4
.............
W
m
angle of
2.0
1.5
incidence
d
50 o
A
d
10
=
30
A
-,
10
10-, 1.0
u
surface level l two tw 0 I.,
10-,
pl=0.5 0.5
....
...........
-6
0.0
I
fl at surface flat Sul s
-2
10-3
10
....
249
2.5
3.0
ir r..
0.0
0.5
angle
[deg]
t
........
1.0
1.5
of incidence
2.0
2.5
3.0
[deg]
Fig. 8.15. Coherent reflectivity of a two level surface calculated within the kinemat5 nm (left) theory for two values of the probability pi’and the step height d 0.5 and for two values of d and a symmetrical probability distribution pi P2
ical
=
=
=
(right)
piroj and P2r,,2. They height d (Fig. 8.15).
give
rise to interference
fringes
which represent the
example of a thin surface layer of porous silicon fits approximately this simple model, if its thickness is smaller than the vertical correlation lengths of the crystallites (Fig. 8.16 (a)) [22]. Since the surface "layer" density is quite different from that of the substrate, we can observe two critical angles 01 and 02. The second one, 02, corresponds to silicon, the first one, 01, to the averaged surface region. Above 01 the wave can penetrate into the perturbed The
region, however total external reflection occurs at the "interface" with the non-perturbed region. That is why very intense fringes appear in this region between 01 and 02, which drop rapidly above 02. The whole curve is similar to that of a homogeneous layer of much less density or to that of a surface grating. In the fitted curves a small Gaussian deviation of the actual displacement around the z, and Z2 has been supposed, which leads to roughness diminution factors of the Fresnel reflection coefficients similar to
surface
(8.24). 8.4.3
Reflection by "Virtual Interfaces" Between Porous Layers
layers are fabricated by electrochemical etching in, a monocrystalline silicon wafer. By a variation of the anode voltage, multilayers of modulated porosity can be produced. Following our division of the layer polarisability we can distinguish between the porous layer volume and the size of the layer of equal porosity. The interface between two layers of,different porosities is not a microscopic laterally continuous and sharp interface between two media of different density, but an interface of two degrees of porosity. According to the coherent approach (used also in Sect. 3.4) we take for the coherent reflection an effective averaged refractive index into account. Porous silicon
Tilo Baumbach and Petr Mikulik
250
(a)
(b)
-2-
-2-
3
-
-3
I
4[
0 0,
01
-4-
.
0.4
0.2
0.0
angle
of incidence
0.6
-5 0.0
0.2
angle
[deg]
0.4 of incidence
0.6
[deg]
(full) and fitted (dashed) reflectivity curves of a thin porous layer (a) and of a porous silicon double layer (b) on silicon substrate [22]. Positions 011 02 are the critical angles of the porous layer and the substrate, respectively Fig.
8.16. Measured
silicon surface
Layers of statistically homogeneous porosity
are
assumed. We treat the slow
of the transition between two
layers of different porosity by a "roughness" function results Same are obtained by introducing a probability transition from of to layer layer. An experimental examporosity graduated for double is in a layer sample [22]. The thickness of the Fig. 8.16(b) ple given surface layer is much smaller than that of the buried layer. The fast oscillating fringes represent the total thickness. The fringe amplitude is modulated by a period, which corresponds approximately to the thickness of the surface layer. It has been found from the simulation that the interface between the two layers of different porosity is much sharper than the interface with the substrate (which is the end front of the etching process). The occurrence of the modulation of thickness oscillations in Fig. 8.16(b) is a direct proof for the validity of the coherent scattering approach. Between the two porous layers there is nowhere a real roughly smooth lateral interface between two media. Nevertheless the x-rays are specularly reflected at this "microscopically non-existent interface" showing all features of the continuum theory of dynamical reflection by multilayers. Gaussian
8.5
Non-Specular X-Ray Reflection
scattering approach (2) within the explicit expressions for the incoherent scattering cross section for x-ray reflection by rough multilayers. We discuss the main features of the scattering patterns illustrated by experimental examples. The representation of the scattering in reciprocal space allows a simple interpretation of the findings by the various scattering processes. We will treat samples with interfaces having a Gaussian roughness profile, diffuse scattering from terraced interfaces and finally non-coplanar diffuse scattering. In this section
DWBA
we
(Chap. 4)
use
the incoherent
and derive
some
8
Interfaces with
8.5.1
X-ray Reflectivity by Rough Multilayers
Gaussian
a
251
Profile
Roughness
having a Gaussian roughness profile. We start with the scattering from a single surface. Then we continue with a multilayer showing the effects of different roughness replication as well as dynamical scattering effects on reciprocal space maps. We will deal with interfaces
Surface
Single
distribution. The case
(see [1,23-25],
for
instance) Z2 + Z/2
1
(Z, Z
P2
Firstly we will deal with surfaces of a gaussian probability pair probability distribution function is in the stationary
27r
0
4
-
exp
C2 ZZ (,rl
2o-2[l
r,
2zz’C
_
-
ZZ
-LC2z(,rll ,4
-
’ril) r111)] (8.38)
with the two-dimensional characteristic function X,, -,,
i(qz-qY))
(q, q’)
e_0,2(q2+q’2)/2 eqq’C_(,r11-r11’)
_
(8.39)
One correlation function, which has been successfully applied to interpret experimental findings, follows from similarities between the description
the
roughness properties and the Brownian motion, position by time. Supposing a behaviour like [1,24]
of interfaces with fractal
replace
we
the lateral
[Z(,rll) leads
together
_
Z(,Pll/)] 2)
=
A
I
Jr11
r11
12h
0 < h
O
from qmi,,
runs
kBT
1:
=
21r/a
where
is
a
qm
kBT _
In
47-y
one
obtains
[I
< 1 and
+
(z 2)
(9.7)
q2
max
=
(71‘49)]
0.4nm
.
(9.8)
Also
interesting
are
limits:
’
For
a
a
In the continuous limit:
z 2) Generally,
A
rigorous calculation
of the statistical average
see
for
example Ref. [5].
the
Jean
284
Daillant
to-,,
2.0
10
10
10
1.5 -30
+
Z 1.0
0.5
10
to
10
-401
0
to,
to,
to, q
Fig.
9.2. left:
to,
to,
10
10
to
-12
10-10
to-,
(m-’)
r
Amplitude
of the
73mN/m (continuous line)
capillary
wave
to-,
10
-4
to-,
(m)
spectrum for surface tension
-y
7.3mN/m (broken line). right: The heightheight correlation function (z(O)z(rjj)) for water (continuous line) and the surface of a liquid having the same density but a 10 times smaller surface tension (long dashed line). The dotted line is obtained by attenuating the spectrum for water at a
molecular cut-off q
and y
=
27r/10-lorn-1
=
A
oo
0
g
(z2) (z 2)
_
_171g,
-
InA.
These
logarithmic divergences do not imply that there is no interface. The exist, but is not localised in space [7], in agreement with the fact that the divergence is due to small wavevector modes q -+ 0". An open question is that of the corrections to the surface energy at very small lengthscales. This is an important question since, as shown by Eq. (9.5) the (deformation) surface energy determines the surface structure. Taking into account the coupling between capillary modes [8] would lead to a renormalisation of the surface tension equivalent to a larger effective surface tension at very small lengthscales. The opposite trend is displayed when we take into account the effect of long-range dispersion forces which lead to a interface does
[9].
smaller effective surface tension The
height-height
correlation function
forming
the spectrum
Eq. (9.5):
g(r1j)
--
Z(o)]2)
=
(z(O)z(rjj))
=
([z(rll)
_
2(Z2
can
be obtained
k.BT _
KO
7r’Y
by Fourier
(ril Izpgl^l)
trans-
(9.9)
and
(figure 9.2 right). Ko is Ko(x).,,o -_ Log2 7, -
kBT
27r-/
Ko
(ril
(9.10)
the modified second kind Bessel -function of order 0. -
Logx and lim Ko(x).,-,,,
=
0.
Reflectivity of Liquid Surfaces and Interfaces
9
285
Chap. 2, there is a specular reflection on a liquid surface only gravity limits the logarithmic divergence. This logarithmic divergence with distance of the roughness, only limited by gravity or the finite size of the surface is a distinctive property of liquid surfaces. Substituting Eq. (9.9) into Eq. (4.41), it is possible to find the following approximation for the scattering cross-section which is valid in the limit q11 >> As indicated in because
lZp_gl-l [10-13]: do-
A
-
dQ
k 04(1
"1
-
and q
area
=
12,
q11 qmax
(kB T/21r7) I q,, 1 12.
Relation to Self-Affine Surfaces
Many solid surfaces
are
Such surfaces
are
known
exponent (see chapter
length.
by correlation functions of the form:
well described
(z(O)z(r11))
tion
(
kBT
’
2 Ito’ll 1 tsc, 0, 12 Tq-
167r2
where A is the illuminated
9.1.2
n2)2
-
.
as
2).
=
0.2e-(ril/ )21’
;0
self-affine surfaces and h is the a
A self-affine fractal differs from
the
and
roughness,
is the
(9.12)
h < 1.
2nM-1. Note also that a data analysis not taking into account diffuse scattering leads to
where F is the
=
This factor is
an erroneous
.
estimation of the structural parameters.
101
5
10-1
4
10-2
3
10-3
2
water
Si
10-4
C
10-1
2
-1
1
0 z
10-
chain
1 2
3
4
(nm)
6
10-7 10-1 10-9 10-
10
0
4
2
q,
6
(nm-1)
Fig. 9.6. Interference pattern resulting from the reflection of an x-ray beam on an octadecyltrichlorosilane monolayer at the air/water interface and its corresponding electron density model (inset, black curve). The broken gurve represents the specular reflection, the long-dashed curve the diffuse intensity, and the thick line the total intensity. The grey curve in the inset is obtained when the data are analysed using a "box model" with error function transition layers, not taking into account diffuse scattering
Jean Daillant
290
Fig.
"Rocking curve" geometry. q,, is approximately kept by "rocking" the incident beam and the detector
9.7.
varied
9.2.3
The
the
Diffuse
case
constant
as
q,
is
Scattering
of diffuse
scattering
is
even
kind of measurements that
more
difficult. One would expect that
successfully carried out on solid surfaces could be also applied to liquid surfaces. In particular, "rocking curves" (equivalent to q__ scans at fixed q,,, Fig. 9.7) would yield a very good resolution 3 along q,: same
/A q,,
27r =
A
sin
are
0i"zA0i" + sin O,C, Osc)
(9.20)
In fact (Fig. 9.8, left) such measurements lead to "flat" spectra for q, 2nm-’. This is because when the incidence angle becomes larger than the
critical
angle for total external reflection, bulk scattering dramatically inlarge q,, (Fig. 9.8, right). This example shows that it is in practice necessary to fix the incidence angle below the grazing angle for total external reflection 0,. It is then possible to measure the scattered intensity either in the plane of incidence (projected on q,) or in the horizontal sample plane. In the first case, q, and q,, are varied together and it is possible to measure the normal structure of, for example, a film, and verify that surface scattering is indeed measured. However, one has to decouple the structural effects from creases
at
the fluctuation
Measuring whenever
one
spectrum.
diffuse
scattering
in the
plane
of incidence should be considered
is interested in the determination of the normal structure of
thin films using synchrotron radiation. This has two main advantages over reflectivity (see Fig. 9.11): The reduced background. The much lighter experimental setup (only a mirror is required instead of
-
-
a
beam
deflector).
The resolution function is in in Refs.
[19,20].
particular discussed in section
4.7.2 of this book and
9
Reflectivity of Liquid Surfaces and Interfaces
10-1
101
5
10-1
2
10-2
107’
10-15
10- --’
11 .
2
10-5
10
10-1 0
10
0.5
0
-0.5
-1.0
Fig. 9.8.
left: Diffuse
2.5nm-’ (dark
i.
-7
1.0
10
101
101
IV qx
scattering (rocking curves)
grey
circles)
-
..
’10 10J
% (x Im-’)
q,
291
and q_
=
101
le
lop
(.-’)
from the bare water surface for
3nm-’ (light
grey
circles)
as a
function
peaks. Calculated surface 2.5nm-’ (black line). A constant background has been added to signal at q, calculate the dashed line, giving a better agreement with the experimental data. COOH) film (CH2)18 Right: Diffuse scattering by an arachidic acid (CH3 at the air/water interface. The fixed angles of incidence are respectively 2mrad the reflected beam is at q.,
of
=
0. Note the Yoneda
=
-
-
(black symbols),
6mrad
2.4mrad. Note that for
(dark-grey symbols) a
grazing angle
and 10mrad
of incidence
(light-grey symbols). O’_
equal
to
sensitivity revealed on the other curves by the constructive 10’m-1 is lost because of bulk scattering
=
10mrad, the surface interference for q.,
"Z
only interested in the roughness spectrum, a second kind of sample plane) which directly yields a signal propor(in tional to the roughness spectrum should be preferred (Fig. 9.10). A last important point which is not specific to liquid surfaces is that the diffuse intensity is proportional to the resolution volume (Fig. 9.12). It is therefore necessary to precisely determine the resolution function as a function of slit openings and of the footprint of the beam on the surface to precisely determine the magnitude of this intensity. When
scan
we are
the horizontal
Jean Daillant
292
Z
q
Nal(TI)
Od X
VacuA "
0
SC
Path
7
Vacuum
Fath F
C*(, 11
S.1C
Langmuir trough Fig. 9.9. Schematics of experiments (Troika beamline, ESRF). (q, q,,) plane of incidence geometry. in-plane q, geometry. C*(111): diamond monochromator, SiC: mirror, Nal(TI): scintillation detector. Typical distances are: Sample-to-Sd distance 700 mm, Sr--Sd distance 500
Sj, S,_ and Sd mm
x
0.250
are wi x
mm.
hi: 0.4
Typical horizontal x
mm
0.2 mm, w,
x
x
vertical
hr-: 2
mm
openings
x
of the slits
2 mm, Wd
x
hd: 10
mm
Y
d
X
Vacuum Path I
P.S.D.
Schematics of experiments (Troika beamline, ESRF). In-plane q, geC*(1-11): diamond monochromator, SiC: mirror, PSD: position sensitive gas-filled (xenon) detector. The experimental curve represents the scattered intensity (horizontal axis) as a function of the vertical position on the PSD. Typical
Fig.9.10.
ometry.
distances izontal mm, w,
are:
x x
Sample-tO-Sd distance 700 MM, Sc-Sd distance 500 mm. Typical horopenings of the slits Si, S,_ and Sd are wi x hj: 0.3 mm x 0.2
vertical
h,-: 0.3
mm
X
100 MM, Wd
x
hd: 0.5
MM
X
100
HIM
9
Reflectivity of Liquid Surfaces and Interfaces
293
10-3
10-4 10-5 10-6 10-7 10-8 10-9
10-101 0
0.25
0.75
0.50
1.00
1.25
10" m-1) q. (x Fig. 9.11. Laboratory (empty circles) and synchrotron (filled circles reflectivity experiments (top). Diffuse scattering experiment in the plane of incidence (filled squares, bottom) for the same arachidic acid monolayer on a CdC12 subphase 100 0 0
10-2
(b
10-4 1
0.9 0.92 0.94
24.5 0
10
20
30
40
50
60
70
80
E*C3
Fig. 9.16. Left: Schematics of the alcohol monolayer at the air/water interface. Top, right: Film thickness as a function of temperature. The arrows indicate the phase transition. Bottom, right: Volume per CH2 as a function of temperature. Note the density jump at the 2-d liquid to solid (rotator phase) transition. With kind permission of B. Berge and J.P. Rieu
the surface pressure,
(CH2)18
-
COOH)
as
illustrated in
Fig. 9.17
for
an
arachidic acid
(CH3
-
film.
order corrections to the spectrum, i.e. effects of the bending stiffof the film are also apparent. Results for a L,, di-palmitoylphosphatidyl-
Higher ness
choline
(DPPC)
film
on
pure water
are
presented
in
Fig.
9.18. Whereas at
small qy values the scattered intensity scales with the surface tension as expected, this is no longer true at large qy due to the effect of bending stiffness.
Fig. 9.18 have been analysed using the spectrum Eq. (9.18) including the additional term Kq’ in the denominator. For the more compressed film of Fig. 9.18 it is found that x (5 2)kBT, smaller than generally expected in condensed DPPC films [34]. The observed wave-vector range is not large enough to allow the precise determination on the exponent 4. Smaller exponen ts are however found with the very rigid films[27] formed by fatty The data of
--
acids (here behenic acid CH3 (CH2)20 COOH) on divalent cation subphases (5 x 10-’mol/I CdC12) at high pH (8.9) and low temperature (5’C). 3.3 Uncompressed, such films exhibit a qpower law which has been attributed to the coupling between in-plane (phonons) and out-of-plane elasticity [27]. Finally, in systems with more than one interface, it is possible to measure -
-
the correlation between the interfaces soap films constants
[35] can
and also for free be measured
[36].
(see
standing ,
Sect.
4.3.3).
This is the
case
for
smectic films for which the elastic
all’
Reflectivity
9
Surfaces and Interfaces
Liquid
299
10
E
tr
0.1 10
E
X
a,?, 0.1
2
5
le
2
q.
5
108
2
5
(m-’)
scattered by an arachidic acid film (black curves) and (grey curves). The surface tensions are (top to bottom) 33 mN/m (diamonds), 43mN/m (triangles), 53mN/m (squares), 69mN/m (circles) and 73mN/m. (b): The same data normalised by 7/7water in order to illustrate the scaling I oc y in the
Fig. 9.17. (a): Intensity
water
range 3 x106,rn
-
1
